Abstract
Let H be a Hilbert space with inner product \((\cdot ,\cdot )_H\) and let \(\varphi \in C^2(H)\). By \(\varphi '(\cdot )\) we denote the Fréchet derivative of \(\varphi \) and by \(\nabla \varphi (\cdot )\) its gradient, that is, \(\nabla \varphi (u)\in H\) for every \(u\in H\) and.
To think freely is great, but to think rightly is greater.
Thomas Thorild, engraved in golden letters at the entrance to the Grand Auditorium, Uppsala University
Let H be a Hilbert space with inner product \((\cdot ,\cdot )_H\) and let \(\varphi \in C^2(H)\). By \(\varphi '(\cdot )\) we denote the Fréchet derivative of \(\varphi \) and by \(\nabla \varphi (\cdot )\) its gradient, that is, \(\nabla \varphi (u)\in H\) for every \(u\in H\) and
where by \(\left\langle \cdot ,\cdot \right\rangle \) we denote the duality brackets for the pair \((H^*, H)\). Recall that \(u_0\) is a critical point of \(\varphi \) if \(\varphi '(u_0)=0\), which by (6.1) is equivalent to saying that \(\nabla \varphi (u_0)=0\). We say that \(c\in \mathbb R\) is a critical level if \(\varphi ^{-1}(c)\) contains critical points. Otherwise \(c\in \mathbb R\) is said to be a regular level.
Let \(a<b\) be two regular values and let \(M=\varphi ^{-1}([a, b])\). The object of Morse theory is the relation between the local topological structure of the level sets of \(\varphi |_{M}\) near a critical point and the topological structure of the manifold M. More precisely, suppose that \(u_0\in H\) is an isolated critical point of \(\varphi \). Then the local behavior of \(\varphi \) near \(u_0\) and the topological type of \(u_0\) are described by a sequence of abelian groups \(\{C_k(\varphi , u_0)\}_{k\in \mathbb N_0}\), known as the “critical groups” of \(\varphi \) at \(u_0\) and defined using homology theory. If the critical point \(u_0\) is nondegenerate (that is, \(\varphi ''(u_0)\in \mathscr {L}(H, H)\) is invertible), then the critical groups can be computed by linearization using the Morse lemma (see Proposition 5.4.19). In fact, if \(u_0\) is nondegenerate, then
with m being the Morse index of \(u_0\) (see Definition 5.4.18(b)). In the degenerate case, no such simple relation exists. Nevertheless, we can still have some results in the degenerate case provided \(\varphi ''(u)\) is a Fredholm operator. The critical groups are invariant under small perturbations of the function \(\varphi \). The global aspect of Morse theory is expressed by the so-called “Morse inequalities”, which relate the critical groups of \(\varphi |_{M}\) to the homology groups \(H_k(\varphi ^b,\varphi ^a)\), \(k\in \mathbb N_0\), which are isomorphic to the homology groups \(H_k(M,\varphi ^{-1}(a))\), \(k\in \mathbb N_0\), by the excision property of homology theory.
Since Morse theory and critical groups use homology theory, in Sect. 6.1 we conduct a quick review of those tools from “Algebraic Topology” which we will use in the sequel. We present all the relevant notions and derive some fundamental consequences of these definitions. Special attention is given to singular homology theory, because this is the homology theory which we will use to define the critical groups, which are defined in Sect. 6.2, and the case of nondegenerate and of degenerate critical points are examined. We also derive the Morse relations which express the global aspects of Morse theory. In Sect. 6.3, we establish the invariance properties of critical groups. So, we show their \(C^{1}\)-invariance and their homotopical invariance. As in degree theory, these properties are very prolific tools in the computation of critical groups of a given functional. In Sects. 6.4 and 6.5, we consider the case of minimizers, maximizers and of saddle points (critical points of mountain pass type). In Sect. 6.6, we introduce homological counterparts of the notions of linking sets (see Definition 5.4.1) and of local linking (see Definition 5.4.14) and compute the critical groups for these more general settings. In Sects. 6.7 and 6.8 we use critical groups to prove the existence of multiple critical points. After all, the importance of critical groups lies in the fact that they provide very efficient tools to generate additional critical points and also to distinguish between critical points.
6.1 Elements of Algebraic Topology
In this section we review some basic definitions and facts of algebraic topology which will be used in the sequel.
Definition 6.1.1
(a) A “pair of spaces” (X, A) is a Hausdorff topological space X together with a subspace \(A\subseteq X\). We write \((X,A)\subseteq (Y, B)\) if \(X\subseteq Y\) and \(A\subseteq B\).
(b) A “map of pairs” (X, A), (Y, B) is a continuous map \(\varphi :X\rightarrow Y\) such that \(\varphi (A)\subseteq B\). We denote the collection of all such maps by
Also, by \(\mathrm{id}_{(X,A)}:(X,A)\rightarrow (X, A)\), we denote the identity map seen as a map of pairs.
(c) A map \(\varphi \) is a “homeomorphism of pairs” (X, A), (Y, B), if \(\varphi :X\rightarrow Y\) is a homeomorphism and \(\varphi ^{-1}\) is a map of pairs (Y, B), (X, A) (that is, \(\varphi |_{A}:A\rightarrow B\) is a homeomorphism and \(\varphi (A)=B\)).
Remark 6.1.2
A space X can be regarded as the pair of spaces \((X,\emptyset )\). If A is a singleton (that is, \(A=\{u_0\}\)), then the pair \((X, A)=(X,\{u_0\})\) is denoted by \((X, u_0)\) and it is usually called a “pointed space”. The composition of two maps of pairs is still a map of pairs.
In Definition 3.1.13 we introduced the notion of homotopy between two continuous maps, which played a central role in degree theory. Sometimes it is necessary to consider homotopies between maps of pairs. Then Definition 3.1.13 is extended easily to the following one:
Definition 6.1.3
Given pairs (X, A) and (Y, B), two maps of pairs \(\varphi ,\psi :(X,A)\rightarrow (Y, B)\) are said to be “homotopic” if there exists a map of pairs
such that
We write \(\varphi \simeq \psi \) to indicate that \(\varphi ,\psi \) are homotopic in the above sense. If \(\varphi ,\psi :(X,A)\rightarrow (Y, B)\) are maps of pairs and \(\varphi |_{A}=\psi |_{A}\), we say that \(\varphi \) and \(\psi \) are “homotopic relative to A” if there exists a homotopy
such that
(that is, the homotopy h is fixed on A). In this case we write
Remark 6.1.4
In the above definition it is said that in continuously deforming \(\varphi \) to \(\psi \), it is required that at each time instant \(t\in [0,1]\), the set A is mapped into B.
Using Definitions 3.1.13 and 6.1.3 we are led to the fundamental notion of “homotopy equivalence” of topological spaces.
Definition 6.1.5
(a) Two Hausdorff topological spaces X, Y are said to be “homotopy equivalent” (or of the “same homotopy type”) if there exist maps \(\varphi :X\rightarrow Y\) and \(\psi :Y\rightarrow X\) such that \(\psi \circ \varphi \) is homotopic to \(\mathrm{id}_X\) and \(\varphi \circ \psi \) is homotopic to \(\mathrm{id}_Y\). In this case the map \(\varphi \) is a “homotopy equivalence” and \(\psi \) is the “homotopy inverse” of \(\varphi \). If X, Y are homotopy equivalent, then we write
(b) Two pairs of spaces (X, A) and (Y, B) are “homotopy equivalent” if there exist maps of pairs \(\varphi :(X,A)\rightarrow (Y, B)\) and \(\psi :(Y,B)\rightarrow (X, A)\) such that \(\psi \circ \varphi \simeq \mathrm{id}_X\) and \(\varphi \circ \psi \simeq \mathrm{id}_Y\) (see Definition 6.1.3; so the homotopies are homotopies of pairs). If the pairs \((X,A),\ (Y, B)\) are homotopy equivalent, then we write
Remark 6.1.6
As the names suggest, both notions are equivalence relations. In general homotopy equivalence of X, Y (respectively of (X, A), (Y, B)), roughly speaking, means that X (respectively (X, A)) can be deformed continuously to Y (respectively (Y, B)). It is easy to see that two homeomorphic spaces are homotopy equivalent but the converse is not true in general. So, the classification of Hausdorff topological spaces up to homeomorphism is more refined than the classification up to homotopy equivalence. Deformation retracts and strong deformation retracts (see Definition 5.3.10(b)) are homotopy equivalences which are easy to visualize. Indeed, if \(A\subseteq X\) is a retract of X, \(r:X\rightarrow A\) is a retraction map (that is, \(r(\cdot )\) is continuous and \(r|_A=\mathrm{id}|_A\)), and \(i:A\rightarrow X\) is the inclusion map, then \(r\circ i=\mathrm{id}_A\) and \(i\circ r\simeq \mathrm{id}_X\) (for deformation retracts) and \(i\circ r\simeq _A\mathrm{id}_X\) (for strong deformation retracts). So, if A is a deformation retract or strong deformation retract of X, then \(A\sim X\).
Using homotopies we can introduce the following fundamental topological notion.
Definition 6.1.7
A Hausdorff topological space X is said to be “contractible” if the identity map \(\mathrm{id}_X:X\rightarrow X\) is homotopic to a constant map \(\varphi :X\rightarrow *\) (that is, there exists continuous map \(h:[0,1]\times X\rightarrow X\) such that \(h(0,u)=u\) for all \(u\in X\) and \(h(1,u)=*\) for all \(u\in X\)).
Remark 6.1.8
Clearly, X is contractible if and only if it is homotopy equivalent to a singleton if and only if every point of X is a deformation retract. A contractible space is simply connected and any two maps into a contractible space are homotopic. Evidently, every convex set of a Banach space or more generally any star-shaped set is contractible (recall that a subset X of a Banach space is star-shaped if there exists a \(u_0\in X\) such that for all \(u\in X\), \([u_0,u]=(1-t)u_0+tu\), \(t\in [0,1]\), lies in X).
Example 6.1.9
(a) Let \(X=S^1=\{u\in \mathbb R^2:|u|=1\}\) (the unit sphere in \(\mathbb R^2\)) and \(Y=S^1\cup [(1,0),(2,0)]\) (recall \([(1,0),(2,0)]=(1-t)(1,0)+t(2,0)\) for all \(t\in [0,1]\)), that is, [(1, 0), (2, 0)] is a closed line segment on the horizontal axis, joining the points (1, 0) and (2, 0). We claim that X and Y are not homeomorphic. Indeed, if from X we remove any point, the remaining set is still connected. On the other hand, if from Y we remove the point (1, 0), the remaining set is disconnected. However, the sets X and Y are homotopy equivalent. To see this, let \(\varphi :X\rightarrow Y\) and \(\psi :Y\rightarrow X\) be the following maps
We have \(\psi \circ \varphi =\mathrm{id}_X\) and \(\varphi \circ \psi \simeq \mathrm{id}_Y\) since \(\varphi \circ \psi =\psi \).
(b) Any convex set in a Euclidean space is homotopy equivalent to a point (just recall that a convex set is contractible, see Remark 6.1.8).
(c) \(S^{N-1}=\{u\in \mathbb R^N:|u|=1\}\ (N\geqslant 2)\) is homotopy equivalent to \(\mathbb R^N\backslash \{0\}\) (just recall that \(S^{N-1}\) is a strong deformation retract of \(\mathbb R^N\backslash \{0\}\)).
(d) Let \(B^2=\{u\in \mathbb R^2:|u|<1\}\) and consider the solid torus \(S^1\times B^2\). This space is homotopy equivalent to \(S^1\). More generally, if V is a vector bundle over a topological space X, the zero section is a strong deformation retract of V, hence homotopy equivalent to it.
Since for our purposes the role of algebraic topology is auxiliary, to avoid a lengthy presentation, we will follow the axiomatic approach of homology theory (naive homology theory).
Definition 6.1.10
Let \(\{G_k\}_{k\in I}\) be a family of abelian groups and \(\{j_k\}_{k\in I}\) a corresponding family of homomorphisms
We say that the sequence (chain) (6.2) is exact if and only if
Remark 6.1.11
If \(G_1,G_2\) are two abelian groups and we consider the chain
then (6.3) is exact if and only if j is an isomorphism.
More generally, suppose that \(G_1,G_2,G_3\) are three abelian groups and consider the following exact chain
From the exactness of (6.4), we see that \(j_1\) is injective and \(j_1(G_1)\) is isomorphic to \(G_1\) and equal to \(\mathrm{ker}\, j_2\). Moreover, \(j_2\) is surjective and we have that \(\mathrm{ker}\, j_2\oplus \mathrm{im}\, j_2\) is isomorphic to \(G_1\oplus G_3\).
Next, we introduce a “homology theory” by listing a number of axioms which must hold. They are usually called the “Eilenberg–Steenrod axioms”.
Definition 6.1.12
A “homology theory” on a family of pairs of spaces (X, A) consists of:
-
(a)
A sequence \(\{H_k(X, A)\}_{k\in \mathbb N_0}\) of abelian groups known as “homology groups” for the pair (X, A) (note that for the pair \((X,\emptyset )\), we write \(H_k(X), k\in \mathbb N_0\)).
-
(b)
To every map of pairs \(\varphi :(X,A)\rightarrow (Y, B)\) is associated a homomorphism
$$\varphi _*:H_k(X,A)\rightarrow H_k(Y, B)\ \text {for all}\ k\in \mathbb N_0.$$ -
(c)
To every \(k\in \mathbb N_0\) and every pair (X, A) is associated a homomorphism
$$\partial :H_k(X, A)\rightarrow H_{k-1}(A)\ \text {for all}\ k\in \mathbb N.$$
These items satisfy the following axioms:
- Axiom 1::
-
If \(\varphi =\mathrm{id}_X\), then \(\varphi _*=\mathrm{id}|_{H_k(X, A)}\).
- Axiom 2::
-
If \(\varphi :(X,A)\rightarrow (Y, B)\) and \(\psi :(Y,B)\rightarrow (Z, C)\) are maps of pairs, then \((\psi \circ \varphi )_*=\psi _*\circ \varphi _*\).
- Axiom 3::
-
If \(\varphi :(X,A)\rightarrow (Y, B)\) is a map of pairs, then \(\partial \circ \varphi _*=(\varphi |_A)_*\circ \partial \).
- Axiom 4::
-
If \(i:A\rightarrow X\) and \(j:(X,\emptyset )\rightarrow (X, A)\) are inclusion maps, then the following sequence is exact
$$\ldots {\mathop {\rightarrow }\limits ^{\partial }}H_k(A){\mathop {\rightarrow }\limits ^{i_*}}H_k(X){\mathop {\rightarrow }\limits ^{j_*}}H_k(X, A){\mathop {\rightarrow }\limits ^{\partial }}H_{k-1}(A)\rightarrow \ldots $$ - Axiom 5::
-
If \(\varphi ,\psi :(X,A)\rightarrow (Y, B)\) are homotopic maps of pairs, then \(\varphi _*=\psi _*\).
- Axiom 6:
-
(Excision): If \(U\subseteq X\) is an open set with \(\bar{U}\subseteq \mathrm{int}\, A\) and \(i:(X\backslash U,A\backslash U)\rightarrow (X, A)\) is the inclusion map, then \(i_*:H_k(X\backslash U,A\backslash U)\rightarrow H_k(X, A)\) is an isomorphism.
- Axiom 7::
-
If \(X=\{*\}\), then \(H_k(\{*\})=0\) for all \(k\in \mathbb N\).
Remark 6.1.13
If an abelian group G is isomorphic to \(H_0(X)\) for every singleton X, then we say that G is the group of coefficients of the homology theory. Note that \(H_k(X, A)=0\) for all \(k\in -\mathbb N\). The excision axiom (see Axiom 6) can be equivalently reformulated as follows:
-
Axiom 6’:
If \(A, B\subseteq X\) and \(X=\mathrm{int}\, A\cup \mathrm{int}\, B\), then the inclusion map \(i:(A,A\cap B)\rightarrow (X, A)\) induces an isomorphism \(i_*:H_k(A,A\cap B)\rightarrow H_k(X, A)\).
Next we derive some useful consequences of the above axioms.
Proposition 6.1.14
If the pairs (X, A) and (Y, B) are homotopy equivalent, then \(H_k(X,A)=H_k(Y, B)\) for all \(k\in \mathbb N_0\) (hereafter, the symbol = denotes that the groups are isomorphic).
Proof
Let \(\varphi :(X,A)\rightarrow (Y, B)\) be a homotopy equivalence and \(\psi \) its homotopy inverse. According to Definition 6.1.5(b) we have \(\psi \circ \varphi \simeq \mathrm{id}_X\). Then Axioms 1 and 2 imply that \(\psi _*\circ \varphi _*=\mathrm{id}_{H_k(X, A)}\). Similarly we show that \(\varphi _*\circ \psi _*=\mathrm{id}_{H_k(Y, B)}\). It follows that \(\varphi _*:H_k(X,A)\rightarrow H_k(Y, B)\) is an isomorphism and \(\varphi ^{-1}_{*}=\psi _*\). \(\square \)
Proposition 6.1.15
If \(A\subseteq X\) is a deformation retract of X, then \(H_k(X, A)=0\) for all \(k\in \mathbb N_0\).
Proof
From Remark 6.1.6 we know that X and A are homotopy equivalent. Hence \(H_k(X)=H_k(A)\) for all \(k\in \mathbb N_0\). Using Axiom 4 we have the exact chain
The exactness of (6.5) and the equality \(H_k(A)=H_k(X)\) for all \(k\in \mathbb N_0\) (that is, \(i_*\) is an isomorphism) imply that \(H_k(X, A)=0\) for all \(k\in \mathbb N_0\). \(\square \)
Corollary 6.1.16
\(H_k(X, X)=0\) for all \(k\in \mathbb N_0\).
Proposition 6.1.17
If A is a retract of X, then \(H_k(X)=H_k(X, A)\oplus H_k(A)\) for all \(k\in \mathbb N_0\).
Proof
Let \(r:X\rightarrow A\) be the retraction and \(i:A\rightarrow X\) the inclusion map. From Definition 3.1.30 we know that \(r\circ i=\mathrm{id}_A\). Then from Axioms 1 and 2, we have
and \(i_*\) is an injection onto a direct summand of \(H_k(X)\). The other summand is the kernel of \(r_*\). Let \(j:(X,\emptyset )\rightarrow (X, A)\) be the inclusion map and consider the sequence
This is an exact sequence (see Axiom 4) and since \(i_*\) is injective, we have \(\mathrm{ker}\, i_*=0\). So, from (6.6) it follows that \(\partial \) is the trivial map. From the exactness of (6.6) it follows that \(j_*\) is surjective. Since \(\mathrm{ker}\, j_*=\mathrm{im}\, i_*\) [see (6.6], \(j_*\) is an isomorphism of \(\mathrm{ker}\, r_*\) onto H(X, A), Therefore we conclude that \(H_k(X)=H_k(X, A)\oplus H_k(A)\) for all \(k\in \mathbb N_0\). \(\square \)
A map of pairs \(\varphi :(X,A)\rightarrow (Y, B)\) defines the maps
Evidently, \(\varphi _2=\varphi |_A\) (see Definition 6.1.1(b)).
We introduce the homomorphisms \(\varphi _*,(\varphi _1)_*\) and \((\varphi _2)_*\) induced by these maps and consider the following diagram:
where \(i,j,i', j'\) are the appropriate inclusions.
Proposition 6.1.18
Diagram (6.7) is commutative.
Proof
We must verify the equalities
The first two equalities follows from Axiom 2 since
The third equality is actually Axiom 3. \(\square \)
To continue we will need an auxiliary result known as the “five lemma”.
Lemma 6.1.19
If we have a commutative diagram of abelian groups and homomorphisms
in which each row is exact and \(\alpha ,\beta ,\delta , e\) are isomorphisms, then \(\gamma \) is an isomorphism.
Proof
First we show that \(\gamma \) is injective. So, suppose that \(\gamma (u_3)=0\) with \(u_3\in G_3\). Then from the commutativity of (6.8), we have
The exactness of the top row in (6.8) and (6.9) imply that we can find \(u_2\in G_2\) such that \(h_2(u_2)=u_3\). Then exploiting once again the commutativity of (6.8), we have
and there exists a \(\hat{u}_1\in \hat{H}_1\) such that
Let \(u_1\in G_1\) such that \(\alpha (u_1)=\hat{u}_1\). We have
But \((h_2\circ h_1)(u_1)=0\) [by the exactness of the row in (6.8)]. Therefore
Next we show that \(\gamma \) is surjective. So, let \(\hat{u}_3\in \hat{G}_3\). Then there is a \(u_4\in G_4\) such that \(\delta (u_4)=\hat{h}_3(\hat{u}_3)\). From the commutativity of (6.8) we have
and the exactness of the lower row in (6.8) implies that
The exactness of the top row in (6.8) implies that there exists a \(u_3\in G_3\) such that
So, there exists a \(\hat{u}_2\in \hat{G}_2\) such that
Let \(u_2\in G_2\) be such that \(\beta (u_2)=\hat{u}_2\). Then
The proof is now complete. \(\square \)
Using this lemma, we can prove the following result.
Proposition 6.1.20
If \((X, A)=\overset{n}{\underset{\mathrm {i=1}}{\bigcup }}(X_i, A_i)\) with \(\{X_i\}^n_{i=1}\) nonempty, closed and pairwise disjoint subsets of X, then \(H_k(X, A)=\overset{n}{\underset{\mathrm {i=1}}{\oplus }}H_k(X_i, A_i)\) for all \(k\in \mathbb N_0.\)
Proof
We do the proof for \(n=2\), the general case following by induction.
Consider the inclusion maps \(i_1:X_1\rightarrow X\) and \(i_2:X_2\rightarrow X\). We show that these maps yield an isomorphism \((i_1)_*\oplus (i_2)_*:H_k(X_1)\oplus H_k(X_2)\rightarrow H_k(X)\) for all \(k\in \mathbb N_0\). To this end it suffices to show that \((i_1)_*,(i_2)_*\) are injective and \(\mathrm{im}\,(i_1)_*\oplus \mathrm{im}\, i_2)_*=H_k(X)\). Let \(j_1:(X,\emptyset )\rightarrow (X, X_1)\) be the inclusion map. Then using Axioms 2 and 6, we have that \((j_1\circ i_2)_*=(j_1)_*\circ (i_2)_*\) is an isomorphism. Hence \((i_2)_*\) is injective and
Similarly we prove that \((i_1)_*\) is injective. Moreover, from Axiom 4 we have
In a similar fashion, we show that the inclusion maps \(i^A_1:A_1\rightarrow A\) and \(i^A_2:A_2\rightarrow A\) produce an isomorphism \(i^A_1\oplus i^A_2:H_k(A_1)\oplus H_k(A_2)\rightarrow H_k(A)\) for all \(k\in \mathbb N_0\). Then for every \(k\in \mathbb N_0\), Axiom 4 gives us a commutative diagram
with the rows being exact and from the previous considerations, we have that \(\alpha ,\beta ,\delta , e\) are all isomorphisms. Invoking Lemma 6.1.19, we infer that \(\gamma \) is an isomorphism too and so we conclude that
The proof is now complete. \(\square \)
Corollary 6.1.21
If \(X={\underset{\mathrm {i\in I}}{\bigcup }}X_i\) is the decomposition of the space into its path components \(X_i\), then \(H_k(X)={\underset{\mathrm {i\in I}}{\oplus }}H_k(X_i)\) for all \(k\in \mathbb N_0\).
Remark 6.1.22
For any Hausdorff topological space \(X, H_0(X)\) is a free abelian group with a basis consisting of an arbitrary point in each path component. Hence \(H_0(X)\) is a direct sum of \(G'\)s, one for each path component of X. If X is path-connected, then \(H_0(X)=G\).
The next proposition generalizes the long exact sequence in Axiom 4.
Proposition 6.1.23
If \(C\subseteq A\subseteq X\) and \(i:(A,C)\rightarrow (X, C)\), \(j:(X,C)\rightarrow (X, A)\), \(\hat{j}:(A,\emptyset )\rightarrow (A, C)\) are the inclusion maps, then the sequence
is exact.
Proof
We first show that
From Axiom 2 it follows that the next diagram is commutative
with \(i_1:A\rightarrow X\) and \(\tilde{j}:(X,\emptyset )\rightarrow (X, C)\) being the inclusion maps. From Axiom 4 we have that \((i_1)_*\circ \partial =0\). Hence \(i_*\circ (\hat{j}_*\circ \partial )=0\) and this proves the first inclusion in (6.12). The other two inclusions in (6.12) are verified in a similar way.
Next we show the opposite inclusions from those in (6.12), namely we show that
Let \(u\in \mathrm{ker}\, j_*\). Using Axioms 2 and 3, we introduce the following commutative diagram of homology groups and homomorphisms
where \(j_1:(X,\emptyset )\rightarrow (X, A)\), \(i_2:C\rightarrow A\) are the inclusion maps and \(\hat{\partial },\tilde{\partial }\) are the boundary maps guaranteed by Definition 6.1.12. The argument is simple and follows the diagram (6.14). It involves four steps:
Step 1: We have \(j_*(u)=0\), hence \((i_2)_*\circ \tilde{\partial }(u)=\hat{\partial }\circ j_*(u)=0\) [see (6.14)]. From the exactness of (6.14), we have \(\mathrm{ker}\,(i_2)_*=\mathrm{im}\,\hat{\partial }\) and so we can find \(y\in H_k(A, C)\) such that \(\hat{\partial }(y)=\tilde{\partial }(u)\).
Step 2: We have \(\tilde{\partial }(i_*(y)-u)=0\) (see (6.14) and Step 1). We know that \(\mathrm{ker}\,\tilde{\partial }=\mathrm{im}\,\tilde{j}_*\). So, we can find \(x\in H_k(X)\) such that \(\tilde{j}_*(x)=i_*(y)-u\).
Step 3: Since \(j_*\circ i_*=0\) (see the second inclusion in (6.12)) and \(j_*(u)=0\) (recall that \(u\in \mathrm{ker}\, j_*\)), we have
But from the exactness of (6.14), we have \(\mathrm{ker}\,(j_1)_*=\mathrm{im}\,(i_1)_*\). So, we can find \(v\in H_k(A)\) such that \(x=(i_1)_*(v)\).
Step 4: From the previous three steps we have
This proves the second inclusion in (6.13). The other two inclusions in (6.13) are proved similarly.
From (6.12) and (6.13), we conclude that the sequence of the proposition is exact. \(\square \)
Corollary 6.1.24
Suppose that \(C\subseteq A\subseteq X\).
-
(a)
If C is a deformation retract of A, then \(H_k(X,A)=H_k(X, C)\) for all \(k\in \mathbb N_0\).
-
(b)
If A is a deformation retract of X, then \(H_k(X,C)=H_k(A, C)\) for all \(k\in \mathbb N_0\).
Next, we focus on homology groups of the form
We start by establishing the precise relation between the homology groups \(H_k(X)\) and \(H_k(X,*)\).
Proposition 6.1.25
\(H_k(X,*)=\mathrm{ker}\, r_*\) where \(r:X\rightarrow \{*\}\) is the map \(r(u)=*\) for all \(u\in X\) and we have \(H_k(X)=H_k(X,*)\oplus H_k(*)\) for all \(k\in \mathbb N_0\).
Proof
We know that \(\{*\}\subseteq X\) is a retract of X. So, from Proposition 6.1.17 we have
The proof is now complete. \(\square \)
Remark 6.1.26
From Axiom 7 and Proposition 6.1.25, we see that
It is often convenient to have a slightly modified version of homology, for which a point has trivial homology groups in all dimensions, including zero. This is done in the next definition.
Definition 6.1.27
The “reduced homology groups” of X are defined by
Remark 6.1.28
Evidently, \(H_0(X)=\tilde{H}_0(X)\oplus G\) and \(H_k(X)=\tilde{H}_k(X)\) for all \(k\in \mathbb N.\)
The next result, known as the “reduced exact homology sequence”, is a particular case of Proposition 6.1.23.
Proposition 6.1.29
If (X, A) is a pair of space and \(*\in A\), then the long sequence of homology groups
is exact.
Reduced homology groups are simple when the space is contractible (see Definition 6.1.7).
Proposition 6.1.30
If X is a contractible Hausdorff topological space, then \(H_k(X,*)=0\) for all \(k\in \mathbb N_0\) and all \(*\in X\).
Proof
From Remark 6.1.8 we know that since X is contractible, every singleton \(\{*\}\) with \(*\in X\) is a deformation retract of X . Invoking Proposition 6.1.15, we have.
The proof is now complete. \(\square \)
Proposition 6.1.31
If \(A\subseteq X\) is a subspace which is contractible in itself, then \(H_k(X, A)=H_k(X,*)\) for all \(*\in A\) and all \(k\in \mathbb N_0\).
Proof
By Propositions 6.1.29 and 6.1.30, we have the following exact chain
The exactness of this chain implies that
The proof is now complete. \(\square \)
The next theorem is a basic tool for computing homology groups. It gives a recipe for computing the homology groups of a space which is the union of two open sets in terms of the homology groups of the two open sets and those of their intersection. This global result is known as the “Mayer–Vietoris theorem”.
We will need the following general result about exact sequences, known in the literature as the “Whitehead–Barratt Lemma”.
Lemma 6.1.32
If the commutative diagram of abelian groups and homomorphisms
has exact rows and \(\gamma _k\) is an isomorphism for all \(k\in \mathbb N_0\), then the sequence
is exact.
The “Mayer–Vietoris theorem” reads as follows.
Theorem 6.1.33
If X is a Hausdorff topological space, \(A, B\subseteq X\) are two nonempty sets whose interiors cover X and \(*\in A\cap B,\) then there is an exact sequence
Proof
From Proposition 6.1.23 we have the following commutative diagram
Here \(i_1,j_1,j_2,\alpha _k,\beta _k,\gamma _k,\alpha _{k-1}\) are the suitable inclusion maps. From Axiom 6’ (see Remark 6.1.13) we know that \(\gamma _k\) is an isomorphism. Then the theorem is a consequence of Lemma 6.1.32. \(\square \)
We will use the previous results to compute the homology groups of the ball \(\bar{B}^n\) and of the sphere \(S^n\) in any homology theory.
So, let
Example 6.1.34
(a) Since \(\bar{B}^n\) is contractible, from Proposition 6.1.30 we have
(b) In contrast \(S^n\) is not contractible. To see this we argue by contradiction. So, suppose that \(S^n\) is contractible. According to Definition 6.1.7 we can find a function \(h\in C([0,1]\times S^n, S^n)\) such that
Using the Tietze extension theorem, we can find \(\hat{h}\in C([0,1]\times \bar{B}^{n+1},\mathbb R^{n+1})\) such that \(\hat{h}|_{[0,1]\times S^n}=h\). We set
From the homotopy invariance of the Brouwer degree (see Proposition 3.1.14), we have
Here by \(u_0\) we mean the constant function \(\psi (u)=u_0\) for all \(u\in S^n\). But from Theorem 3.1.25(a), we have
On the other hand since \(u_0\in S^n\), we have \(d(u_0,B^{n+1}, 0)=0\), a contradiction. This proves that \(S^n\) is not contractible (see also Proposition 3.1.32).
So, to compute the reduced homology groups of \(S^n\), we proceed as follows.
First note that the homology groups \(H_k(S^n,*)\) depend only on the homotopy type of \(S^n\) (see Proposition 6.1.14). So, without any loss of generality we can take the Euclidian norm on \(\mathbb R^{n+1}\).
If \(n=0\), then from Corollary 6.1.16 and Proposition 6.1.17, we have
Now let \(n\geqslant 1\) and let \(u_N\in S^n\) and \(u_S\in S^n\) be the north and south poles respectively. Set
Then \(S^n=S^n_1\cup S^n_2\) and so by Theorem 6.1.33, we have the exact sequence
Note that the spaces \(S^n_1,S^n_2\) are contractible. Therefore
Also, the pair \((S^n_1\cap S^n_2,*)\) is clearly homotopically equivalent to \((S^{n-1},*)\) and so, by Proposition 6.1.14
From (6.16), (6.17), (6.18) we obtain
Then by induction we have
So, in any homology theory, \(H_n(S^n,*)\) is the only reduced homology group of \(S^n\) which is nontrivial and it coincides with \(H_0(*)\) (the group of coefficients of the homology theory).
(c) From the previous two examples and the reduced exact homology sequence (see Proposition 6.1.29), we have
Remark 6.1.35
If X is an infinite-dimensional Banach space and \(\partial B_1=\{u\in X:||u||=1\}\), then \(\partial B_1\) is contractible (compare with Example 6.1.34(b)).
Proposition 6.1.36
If \(X_1\subseteq \cdots \subseteq X_{k+1}\) are Hausdorff topological spaces, then \(\mathrm{rank}\, H_n(X_{k+1}, X_1)\leqslant \overset{k}{\underset{\mathrm {i=1}}{\sum }}\mathrm{rank}\, H_n(X_{i+1}, X_i)\) for all \(n\in \mathbb N_0\).
Proof
Consider the triple \((X_{k+1},X_k, X_1)\) and the long exact sequence corresponding to it according to Proposition 6.1.23. We have
Then from the rank theorem we have
Since for \(k=1\), the inequality claimed by the proposition is in fact an equality the result follows from (6.20) and induction on \(k\in \mathbb N\). \(\square \)
Proposition 6.1.37
If \(X_1\subseteq X_2\subseteq X_3\subseteq X_4\) are Hausdorff topological spaces then for all \(n\in \mathbb N\),
Proof
We consider the triple \((X_3,X_2,X_1)\) and the long exact sequence corresponding to it according to Proposition 6.1.23. We have
From the rank theorem we have
Similarly, if we consider the triple \((X_4,X_3,X_1)\), then from Proposition 6.1.23 we have the long exact sequence
As above, via the rank theorem and the exactness of (6.23), we obtain
Using (6.22) in (6.24), we obtain
The proof is now complete. \(\square \)
Next we introduce a concrete homology theory which we will use in the sequel and which is known as “singular homology theory”.
Singular homology theory extended simplicial homology theory to general topological spaces. Simplicial homology theory is defined for a special kind of spaces, namely compact polyhedra and the complexes resulting from them.
Let \(\Delta _n\) be the standard n-simplex defined by
For \(k\in \{0,\ldots , n\}\) we set \(e_k=(0,\ldots , 0,1,0,\ldots , 0)\) (with 1 located at the \(k+1\)-entry).
Definition 6.1.38
Let X be a Hausdorff topological space.
-
(a)
A “singular n-simplex” is a continuous map \(\sigma :\Delta _n\rightarrow X\).
-
(b)
The free abelian group with the singular n-simplexes as generators and coefficients in \(\mathbb Z\) is called the “\(n \,{\mathop {=}\limits ^{th}}\) singular chain group” and is denoted by \(C_n(X)\). For \(n<0,\ C_n(X)=0\) and if \(c\in C_n(X)\), then c is called a “singular n-chain”.
Remark 6.1.39
The world “singular” is used here to reflect the fact that the map \(\sigma (\cdot )\) need not be a homeomorphism and can have “singularities”. Moreover, its image \(\sigma (\Delta _n)\) in general does not look at all like a simplex. A singular 0-simplex is a map from the singleton \(\Delta _0\) into X. Hence it can be identified with a point of X. A singular 1-simplex is a continuous map \(\sigma :\Delta _1\simeq [0,1]\rightarrow X\), hence it is a path in X.
Definition 6.1.40
(a) Let X, Y be Hausdorff topological space and \(\varphi :X\rightarrow Y\) a continuous map. If \(\sigma :\Delta _n\rightarrow X\) is a singular n-simplex in X, then the composition \(\varphi \circ \sigma :\Delta _n\rightarrow Y\) is a singular n-simplex in Y, denoted by \(\varphi \sigma \). Suppose that \(c=\overset{n}{\underset{\mathrm {k=1}}{\sum }}a_k\sigma _k\), where \(a_k\in \mathbb Z\) is an n-chain X (that is, \(c\in C_n(X)\)). Then
and the homomorphism \(\varphi _*:C_n(X)\rightarrow C_n(Y)\) is the “homomorphism induced by \(\varphi \)”.
(b) For each \(k\in \{0,\ldots , n\}\), let \(d_k:\Delta _{n-1}\rightarrow \Delta _n\) be the affine function defined by
is called the “k-face function in dimension n”. For every singular n-simplex \(\sigma :\Delta _n\rightarrow X\), the “boundary of \(\sigma \)” is defined to be the singular \((n-1)\)-chain \(\partial \sigma \) defined by
This extends uniquely to a homomorphism \(\partial :S_n(X)\rightarrow S_{n-1}(X)\) known as the “boundary operator”.
Remark 6.1.41
Sometimes we write \(\partial _n\) instead of \(\partial \) in order to indicate the chain group on which the boundary operator is acting.
The next proposition gives the most important feature of the boundary operator. Its proof is straightforward but it involves a tedious calculation and so it is omitted.
Proposition 6.1.42
\(\partial _n\circ \partial _{n+1}=0\ \text {for all}\ n\in \mathbb N_0.\)
Definition 6.1.43
(a) A singular n-chain c is said to be an “n-cycle” if \(\partial c=0\).
(b) A singular n-chain c is said to be an “n-boundary” if there is an \((n+1)\)-chain b such that \(\partial b=c\).
(c) By \(Z_n(X)\) we denote the set of all n-cycles and by \(B_n(X)\) the set of all n-boundaries. Both are abelian subgroups of \(C_n(X)\).
Example 6.1.44
(a) Recall that a singular 1-simplex is a path \(\sigma :[0,1]\rightarrow X\) and \(\partial \sigma \) corresponds to the formal difference \(\sigma (1)-\sigma (0)\). Hence a 1-cycle is a formal \(\mathbb Z\)-linear combination of paths with the property that the set of initial points counted with multiplicities is the same as the set of terminal points with multiplicities.
(b) In the case of a singular 2-simplex \(\sigma :\Delta _2\rightarrow X\), the boundary is the sum of three paths with signs. Consider \(\sigma :i_{\Delta ^2}:\Delta ^2\rightarrow \mathbb R^3\) the inclusion map. Then
So, \(\partial i_{\Delta _2}\) is the sum of the singular 1-simplexes in the boundary of \(\Delta _2\) with appropriate signs.
Now we can define the singular homology groups.
Definition 6.1.45
Let X be a Hausdorff topological space. The collection \(\{C_n(X),\partial _n\}_{n\geqslant 0}\) is called a “singular chain complex for X”. We set
Both are abelian subgroups of \(C_n(X)\) and by Proposition 6.1.42 we have
So, we can define the quotient groups
This is the “n-th singular homology group of X”. The singular homology of X is the collection
Remark 6.1.46
The elements of \(H_n(X)\) are called singular homology classes, the coset \(u+B_n(X)\) being the class for the singular n-cycle u. Two n-cycles u and \(u'\) are said to be homologous if they belong to the same singular homology class. Clearly, u and \(u'\) are homologous if and only if \(u-u'=\partial _{n+1}c\) for some singular \((n+1)\)-chain c. If \(H_n(X)\) is finitely generated, then \(\mathrm{rank}\, H_n(X)\)=the n-th Betti number of X. Since \(Z_n(X), B_n(X)\) are subgroups of the abelian group \(S_n(X)\), they are normal subgroups.
We can also define relative singular homology groups.
Definition 6.1.47
Let (X, A) be a pair of spaces. We set
This is the “relative n-singular chain group of X mod A”, which is a free abelian group with generators those singular n-simplexes \(\sigma :\Delta _n\rightarrow X\) whose images are not completely contained in A. The elements of \(C_n(X, A)\) are called “relative singular n-chains of X mod A”. Because \(\partial _n:C_n(X)\rightarrow C_{n-1}(X)(n\in \mathbb N)\) is a homomorphism and \(\partial _n(C_n(A))\subseteq C_{n-1}(A)\), there exists a unique homomorphism
(for notational economy we use the same symbol). This is the “boundary operator” for the relative singular homology groups. As before (see Proposition 6.1.42), we have
We set
(the subgroup of relative singular n-cycles of X mod A),
(the subgroup of relative singular n-boundaries of X mod A).
We have \(B_n(X,A)\subseteq Z_n(X, A)\) and so we can define
This is the “n-th relative singular homology group of X mod A”. This is a free abelian group and if it is finitely generated, then \(\mathrm{rank}\, H_n(X, A)\) is the “n-th Betti number” of the pair (X, A).
Remark 6.1.48
We have
and \(B_n(X, A)=B_n(X)+C_n(A)\) (that is, the subgroup generated by \(B_n(X)\) and \(C_n(A)\)). If \(A=\emptyset \), then \(H_n(X,\emptyset )=H_n(X)\).
Proposition 6.1.49
The relative singular homology introduces a homology theory in the sense of Definition 6.1.12 on the collection of all pairs of spaces.
Remark 6.1.50
We have defined singular homology theory using \(\mathbb Z\) as the group of coefficients, because this is the most standard singular homology. However, in some cases, in order to avoid torsion phenomena, we replace \(\mathbb Z\) by a field \(\mathbb {F}\). In this case \(H_n(X, A)\), \(n\in \mathbb N_0\), is a vector space. Recall that in the presence of torsion, we may have \(\mathrm{rank}\, H_n(X, A)=0\) although \(H_n(X, A)\ne 0\). Finally, we mention that \(\mathrm{rank}\, H_0(X)\) coincides with the number of path components of X. More generally, if \(A\subseteq X\), then \(\mathrm{rank}\, H_0(X, A)\) coincides with the number of path components \(C\subseteq X\) which do not intersect A. So, if each \(u\in X\) can be connected to an element of A by a path in X, then \(H_0(X, A)=0\).
6.2 Critical Groups, Morse Relations
In Morse theory the local behavior of a smooth function \(\varphi \) near an isolated critical point is described by a sequence of abelian groups, known as “critical groups”.
So, let X be a Banach space, \(\varphi \in C^1(X)\) and \(c\in \mathbb R\). From Sects. 5.2 and 5.3 we recall the following notation:
Definition 6.2.1
Suppose that \(u\in K_{\varphi }\) is isolated. The “critical groups” of \(\varphi \) at u are defined by
where \(H_*\) denotes the relative singular homology group with \(\mathbb Z\) being the group of coefficients and U is a neighborhood of u such that \(K_{\varphi }\cap U=\{u\}\).
Remark 6.2.2
The excision property (see Definition 6.1.12, Axiom 6) implies that the above definition is independent of the choice of the neighborhood U for which we have \(K_{\varphi }\cap U=\{u\}\) (recall u is isolated). If we choose the elements of a field \(\mathbb {F}\) as coefficients for the homology groups, then the critical groups are \(\mathbb {F}\)-vector spaces. From the above definition it is clear that the critical groups depend only on the behavior of \(\varphi \) near u. Evidently, they can be defined, even if \(\varphi \) is defined only in a neighborhood of u. This will become even more evident in the next section. Finally, recall that \(C_k(\varphi , u)=0\) for all \(k\in -\mathbb N\).
Proposition 6.2.3
If \(u\in X\) is a local minimizer of \(\varphi \in C^1(X)\) which is an isolated critical point, then \(C_k(\varphi ,u)=\delta _{k, 0}\mathbb Z\) (recall \(\delta _{k, m}=\left\{ \begin{array}{ll}1&{}\text {if}\ k=m\\ 0&{}\text {if}\ k\ne m\end{array}\right. \) for all \(k, m\in \mathbb N_0\), the “Kronecker symbol”).
Proof
Since u is a local minimizer and an isolated critical point of \(\varphi \), we can find a neighborhood U of u such that
Therefore according to Definition 6.2.1, we have
The proof is now complete. \(\square \)
The situation is more involved with local maximizers.
Proposition 6.2.4
If \(u\in X\) is a local maximizer of \(\varphi \in C^1(X)\) which is an isolated critical point, then when \(\mathrm{dim}\, X=m<\infty \), we have \(C_k(\varphi ,u)=\delta _{k, m}\mathbb Z\) for all \(k\in \mathbb N_0\), when \(\mathrm{dim}\, X=\infty \), we have \(C_k(\varphi , u)=0\) for all \(k\in \mathbb N_0\).
Proof
Since u is a local maximizer and an isolated critical point of \(\varphi \), we can find \(r>0\) such that
Here \(\bar{B}_r(u)=\{y\in X:||y-u||\leqslant r\}\).
First assume that \(\mathrm{dim}\, X=m<\infty \). Consider the deformation
Evidently, we have
with \(\partial B_r(u)=\{y\in X:||y-u||=r\}\). Therefore \(\partial B_r(u)\) is a deformation retract of \(\bar{B}_r(u)\backslash \{u\}\) (see Definition 5.3.10(b)). Then Definition 6.2.1 and Corollary 6.1.24(a), together with (6.26), imply that
But from Example 6.1.34(c) we know that
Next, assume that \(\mathrm{dim}\, X=\infty \). In this case we know that both \(\bar{B}_r(u)\) and \(\bar{B}_r(u)\backslash \{u\}\) are contractible (see Remark 6.1.35) and so from Proposition 6.1.30 and 6.1.31, we conclude that
The proof is now complete. \(\square \)
When \(X=\mathbb R\), the critical groups at any isolated critical point can be completely described.
Proposition 6.2.5
If \(X=\mathbb R,\ \varphi \in C^1(\mathbb R)\) and \(u\in K_{\varphi }\) is isolated, then one of the following three situations can occur:
-
(a)
if u is a local minimizer of \(\varphi \), then \(C_k(\varphi ,u)=\delta _{k, 0}\mathbb Z\) for all \(k\in \mathbb N_0\);
-
(b)
if u is a local maximizer of \(\varphi \), then \(C_k(\varphi ,u)=\delta _{k, 1}\mathbb Z\) for all \(k\in \mathbb N_0\);
-
(c)
for all other cases \(C_k(\varphi , u)=0\) for all \(k\in \mathbb N_0\).
Proof
Evidently, (a) and (b) follow from Propositions 6.2.3 and 6.2.4, respectively.
Since u is isolated, we can find \(\epsilon >0\) such that
By hypothesis u is not a local extremum of \(\varphi \). So, \(\varphi \) is either increasing or decreasing on \([u-\epsilon , u+\epsilon ]\). To fix things, we assume that \(\varphi \) is increasing (the reasoning in the same if \(\varphi \) is decreasing). Then
The proof is now complete. \(\square \)
Now we pass to a Hilbert space setting, where Morse theory is more effective. So, let \(X=H=\) a Hilbert space and \(\varphi \in C^2(H)\). Recall that \(u\in K_{\varphi }\) is “nondegenerate” if the self-adjoint operator \(\varphi ''(u)\in \mathscr {L}(H, H)\) is invertible. The dimension of the negative space of \(\varphi ''(u)\) is called the “Morse index of u” and is denoted by \(m=m(u)\in \mathbb Z\cup \{+\infty \}\) (see Definition 5.4.18). Using the Morse lemma (see Proposition 5.4.19), we can compute the critical groups of \(\varphi \in C^2(H)\) at an isolated critical point u which is nondegenerate.
Proposition 6.2.6
If H is a Hilbert space, \(\varphi \in C^2(H)\) and \(u\in K_{\varphi }\) is isolated and nondegenerate, with Morse index m, then \(C_k(\varphi ,u)=\delta _{k, m}\mathbb Z\) for all \(k\in \mathbb N_0\).
Proof
By replacing \(\varphi \) with \(\psi (u)=\varphi (u+y)-\varphi (u)\) for all \(y\in H\) if necessary, we see that without any loss of generality we may assume that
Invoking Proposition 5.4.19 (the Morse lemma), we can find a Lipschitz continuous homeomorphism of a neighborhood W of 0 onto a neighborhood U of 0 such that
(here, by \((\cdot ,\cdot )_H\) we denote the inner product of H). If \(\psi (u)=\varphi (h(u))\) and \(B\subseteq W\) is a closed ball centered at 0, we have
Since u is nondegenerate, \(\varphi ''(0)\) is invertible and so we have
with \(\psi ''(0)\) positive (respectively negative) definite on \(H_+\) (respectively \(H_-\)). So, any \(v\in U\) can be decomposed in a unique way as \(v=v_-+v_+\) with \(v_-\in H_-, v_+\in H_+\). We consider the deformation \(\xi :[0,1]\times B\rightarrow B\) defined by
Then from (6.28) and exploiting the orthogonality of the component spaces, we have
This shows that \(V_-\cap B\backslash \{0\}\) is a deformation retract of \(\psi ^{\circ }\cap B\backslash \{0\}\) and \(V_-\cap B\) is a deformation retract of \(\psi ^{\circ }\cap B\). Then we have
The proof is now complete. \(\square \)
What about degenerate critical points? For this case we have the so-called “Shifting Theorem”, which says that for a degenerate critical point, the critical groups depend on the Morse index and on the “degenerate part” of the functional. Thus the computation of the critical groups is reduced to a finite-dimensional problem. The Shifting Theorem will be proved with the help of an extension of the Morse lemma (see Proposition 5.4.19), which we prove first.
We start with a definition.
Definition 6.2.7
Let X and Y be two Banach spaces and let \(L\in \mathscr {L}(X, Y)\). We say that L is a “Fredholm operator” if \(\mathrm{ker}\, L\) is finite-dimensional and \(R(L)=L(X)\) is finite codimensional (that is, \(\mathrm{dim}\,(Y/\mathrm{ker}\, L)<\infty \)). The number
is called the “index” of L. The set of all Fredholm operators \(L:X\rightarrow Y\) is denoted by \(\mathrm{Fred}\,(X, Y)\).
Remark 6.2.8
If \(L\in \mathrm{Fred}\,(X, Y)\), then \(R(L)\subseteq Y\) is closed. Moreover, we have
and \(L|_V\) is an isomorphism of V onto L(X). The set \(\mathrm{Fred}\,(X, Y)\) is open in \(\mathscr {L}(X, Y)\) and the map \(L\rightarrow i(L)\) is continuous (hence, it is constant on each connected component of \(\mathrm{Fred}\,(X, Y)\)). Every \(L\in \mathrm{Fred}\,(X, Y)\) is invertible modulo finite rank operators, that is, there exists an \(S\in \mathscr {L}(Y, X)\) such that both
are finite rank operators. Finally, if \(K\in \mathscr {L}_c(X, X)\), then \(\lambda \mathrm{id}_X-K\) is a Fredholm operator for every \(\lambda \ne 0\).
Next we state and prove an extension of the Morse lemma which we will need in the proof of the shifting theorem.
Proposition 6.2.9
If H is a Hilbert space, U is an open neighborhood of the origin \(\varphi \in C^2(U),\ 0\in K_{\varphi }\) with \(\mathrm{dim ker}\, \varphi ''(0)>0\), \(L=\varphi ''(0)\) is a Fredholm operator hence
and so every \(u\in H\) admits a unique decomposition
then there exists an open neighborhood V of the origin, an open neighborhood W of the origin in \(\mathrm{ker}\, L\), a homeomorphism h from V into U and a function \(\hat{\varphi }\in C^2(W)\) such that
and \(\varphi (h(u))=\frac{1}{2}(Lv, v)_H+\hat{\varphi }(w)\) for all \(u\in V\).
Proof
Let \(P \in \mathscr {L}(H, H)\) be the orthogonal projection onto R(L). The implicit function theorem implies that we can find \(\rho _1>0\) and a \(C^1\)-function
such that \(\sigma (0)=0,\sigma '(0)=0\) and
We let \(W=B_{\rho _1}\cap \mathrm{ker}\, L\) (an open neighborhood of the origin in \(\mathrm{ker}\, L\)) and consider \(\hat{\varphi }:W\rightarrow \mathbb R\) defined by
Evidently, \(\hat{\varphi }\in C^1(W)\) and using (6.30) we have
So, we have
On \([0,1]\times U\), we define the function
and the vector field
We consider the following abstract Cauchy problem
We will establish the existence of a local flow for (6.31). To this end let
Then using (6.30) we see that
It follows that
From (6.32) and (6.33) we infer that for every \(\epsilon >0\), there exists a \(\delta (\epsilon )\in (0,\rho _1)\) such that
Recall that \(L|_{R(L)}\) is invertible. So, we can find \(c>0\) such that
For \(v\ne 0\), we have
Let \(\epsilon =\frac{1}{2c}\). Using (6.34) and (6.35), we see that
By definition \(g(t, 0,w)=0\) and so we see that g is continuous. Let \(\rho \in (0,\delta (\epsilon ))\) be such that
Using (6.34), (6.35) and (6.37), we see that we can find \(c_1>0\) such that
Now from (6.36) and the mean value theorem, we see that we can find \(c_2>0\) such that
So, the flow \(\gamma (\cdot )\) of (6.31) exists locally. Since \(\gamma (t, 0,w)=0\), the flow \(\gamma \) is well-defined on \([0,1]\times V\) with V a neighborhood of the origin in H. We set
The invertibility of h follows from the invertibility of the flow \(\gamma (1,\cdot , w)\). Then h is the desired local homeomorphism. \(\square \)
To prove the shifting theorem, we will need one more auxiliary result. First a definition.
Definition 6.2.10
For a Hausdorff topological space X, the quotient space
is called the “suspension of X” or “double cone over X”.
Remark 6.2.11
So, the suspension \(\Sigma X\) of X is obtained from \([-1,1]\times X\) by identifying each of the subsets \(\{-1\}\times X\) and \(\{+1\}\times X\) with two different points. The following figure explains this notion and justifies its name
Proposition 6.2.12
If \(A\subseteq \mathbb R^n\), \(0\in A\) and \(B^m\) is the m-ball, then\(H_k(B^m\times A,(B^m\times A)\backslash \{0\})=H_{n-m}(A, A\backslash \{0\})\).
Proof
Let \(m\geqslant 2\) and recall that \(B^m\) is homeomorphic to \([-1,1]^m\). Then we have
Then the result follows by induction from the case \(m=1\).
From the excision property, we have
We introduce the sets
where \(u_+\) and \(u_-\) are the two points which are identified with \(\{1\}\times A\) and \(\{-1\}\times A\) respectively. Also, set
We have
Evidently,
So, we have
Using Theorem 6.1.33 (the Mayer–Vietoris theorem), we obtain
This proves the proposition for \(m=1\) and then by induction for every \(m\in \mathbb N\). \(\square \)
Now we are ready to state and prove the shifting theorem, which takes care of the degenerate case.
Theorem 6.2.13
If H is a Hilbert space, \(U\subseteq H\) is open, \(\varphi \in C^2(U)\) and \(u\in K_{\varphi }\) is isolated with finite Morse index m and \(\mathrm{dim ker}\,\varphi ''(u)\) is finite too, then \(C_k(\varphi ,u)=C_{k-m}(\hat{\varphi }, 0)\) for all \(k\in \mathbb N_0\), with \(\hat{\varphi }\) as in Proposition 6.2.9.
Proof
Without any loss of generality, we may assume that \(u=0\). Let \(C\subseteq U\) be a closed neighborhood of the origin. Using Proposition 6.2.9, we set
(we have kept the notation introduced in Proposition 6.2.9). We have
By hypothesis, \(0\in \mathrm{ker}\, L\) is the only critical point of \(\hat{\varphi }\in C^2(W)\). Since \(\mathrm{dim ker}\, L\) is finite, the Palais–Smale condition is satisfied over any closed ball \(B_r\subseteq W\). From the deformation theorem (see Theorem 5.3.7), we can find \(\epsilon >0\) and \(E\subseteq W\) closed positively invariant for the negative gradient flow such that \(\hat{\varphi }^c\cap E\) is a strong deformation retract of \(\hat{\varphi }^{c+\epsilon }\cap E\) and \(\hat{\varphi }\) is nondecreasing along this deformation h. We set
(see the proof of Proposition 6.2.6). We can easily check that \(H_-\cap \hat{\varphi }^c\cap E\) is a strong deformation retract of \(\psi ^c\cap C\) and \((H_-\times (\hat{\varphi }^c\cap E))\backslash \{0\}\) is a strong deformation retract of \((\psi ^c\cap C)\backslash \{0\}\). Therefore we have
If \(m=\mathrm{dim}\, H_-=0\), then
which is the result of the theorem.
If \(m=\mathrm{dim}\, H_-\geqslant 1\), then using Proposition 6.2.12 we have
The proof is now complete.\(\square \)
Let \(m^*(u)=m(u)+\mathrm{dim\, ker}\, L\) (the extended Morse index of u). Then from Theorem 6.2.13 we infer the following result.
Corollary 6.2.14
If everything is as in Theorem 6.2.13 and \(C_k(\varphi , u)\ne 0\), then \(m(u)\leqslant k\leqslant m^*(u)\).
We return to the more general setting of a Banach space X. We show that nontrivial singular homology groups imply the presence of a critical level between two levels \(a<b\). More precisely, we have the following property.
Proposition 6.2.15
If X is a Banach space, \(\varphi \in C^1(X)\) satisfies the C-condition and there exist \(k_0\in \mathbb N_0\) and levels \(a, b\in \mathbb R\) such that \(a<b\) and
then \(K_{\varphi }\cap \varphi ^{-1}([a, b])\ne \emptyset \).
Proof
We argue indirectly. So, suppose that \(K_{\varphi }\cap \varphi ^{-1}([a, b])=\emptyset \). Then Corollary 5.3.13 implies that \(\varphi ^a\) is a strong deformation retract of \(\varphi ^b\). Proposition 6.1.15 implies that
a contradiction to our hypothesis that \(H_{k_0}(\varphi ^b,\varphi ^a)\ne 0\). \(\square \)
We can be more precise and relate the change in the topology of sublevel sets across a critical level to the critical groups of the critical points for that level.
Proposition 6.2.16
If X is a Banach space, \(\varphi \in C^1(X),a, b\in \mathbb R\) with \(a<b,\varphi \) satisfies the \(C_{c'}\)-condition at every level \(c'\in \left[ a,b\right) ,K_{\varphi }\cap [a, b]=\{c\}\) with \(c\notin \{a, b\}\) and \(K^c_{\varphi }=\{u_i\}^n_{i=1}\) is finite, then \(H_k(\varphi ^b,\varphi ^a)=\overset{n}{\underset{\mathrm {i=1}}{\oplus }}C_k(\varphi , u_i)\) for all \(k\in \mathbb N_0\); in particular
Proof
Using Corollary 5.3.13 and Corollary 6.1.24, we see that
Let \(\{U_i\}^n_{i=1}\) be pairwise disjoint open neighborhoods of the critical points \(\{u_i\}^n_{i=1}\) such that
We have that \(U_i\cap K_{\varphi }=\{u_i\}\) and so from Definition 6.2.1 it follows that
From the excision property of singular homology (see Definition 6.1.12 and Proposition 6.1.49) and using Proposition 6.1.20, we obtain
From (6.42) and (6.43) we conclude that
In particular, from the above isomorphism, we infer that
The proof is now complete. \(\square \)
What can be said about the change in the topology when we cross multiple critical levels? In this direction, we have the following result.
Proposition 6.2.17
If X is a Banach space, \(\varphi \in C^1(X),\ -\infty<a<b<\infty \) are regular values of \(\varphi \), \(\varphi ^{-1}([a, b])\cap K_{\varphi }\) is finite and \(\varphi \) satisfies the \(C_c\)-condition for every \(c\in [a, b]\), then \(\mathrm{rank}\, H_k(\varphi ^b,\varphi ^a)\leqslant {\underset{\mathrm {u\in K^{[a,b]}_{\varphi }}}{\sum }}\mathrm{rank}\, C_k(\varphi , u)\) where \(K^{[a, b]}_{\varphi }=\varphi ^{-1}([a, b])\cap K_{\varphi }\).
Proof
Let \(\{c_i\}^n_{i=1}\) be the critical values of \(\varphi \) in (a, b) in increasing order (that is, \(c_1<\ldots < c_n\)). Let \(\{a_i\}^{n+1}_{i=1}\subseteq [a, b]\) be such that
Using Proposition 6.1.36 with \(X_i=\varphi ^{a_i}\) we have
The proof is now complete.\(\square \)
We can make the above result more precise, with the so-called “Morse relation”. The next definition introduces some algebraic quantities which are important in this direction.
Definition 6.2.18
Let X be a Banach space, \(\varphi \in C^1(X), a, b\in \mathbb R\backslash \varphi (K_{\varphi })\), \(a<b\), and suppose that \(\varphi ^{-1}((a, b))\) contains a finite number of critical points \(\{u_i\}^n_{i=1}\).
-
(a)
The “Morse-type numbers” of \(\varphi \) for (a, b) are defined by
$$M_k(a, b)=\overset{n}{\underset{\mathrm {i=1}}{\sum }}\mathrm{rank}\, C_k(\varphi , u_i)\ \text {for all}\ k\in \mathbb N_0.$$Suppose that \(M_k(a, b)\) is finite for every \(k\in \mathbb N_0\) and vanishes for all large \(k\in \mathbb N_0\). We define
$$M(a, b)(t)={\underset{\mathrm {k\geqslant 0}}{\sum }}M_k(a, b)t^k\ \text {for all}\ t\in \mathbb R.$$Then \(M(a, b)(\cdot )\) is called the “Morse polynomial” of \(\varphi \) for (a, b).
-
(b)
The “Betti-type numbers” of \(\varphi \) for (a, b) are defined by
$$\beta _k(a, b)=\mathrm{rank}\, H_k(\varphi ^b,\varphi ^a)\ \text {for all}\ k\in \mathbb N_0.$$Suppose that \(\beta _k(a, b)\) is finite for all \(k\in \mathbb N_0\) and vanishes for all large \(k\in \mathbb N_0\).
We define
$$P(a, b)(t)={\underset{\mathrm {k\geqslant 0}}{\sum }}\beta _k(a, b)t^k\ \text {for all}\ t\in \mathbb R.$$Then \(P(a, b)(\cdot )\) is called the “Poincaré polynomial” of \(\varphi \) for (a, b).
To prove the “Morse relation”, we will need the following simple lemma.
Lemma 6.2.19
If \(D_0\subseteq D_1\subseteq \ldots \subseteq D_n(n\geqslant 2)\) are Hausdorff topological spaces and \(\mathrm{rank}\, H_k(D_i, D_{i-1})\) is finite for all \(k\in \mathbb N_0\) and all \(i\in \{1,\ldots , n\}\), and vanishes for all large \(k\in \mathbb N_0\), then
where Q(t) is a polynomial with nonnegative integer coefficients.
Proof
We prove the statement for \(n=2\), the general case following by induction. For the triple \((A_2,A_1,A_0)\) we consider the corresponding long exact sequence. We have
From Proposition 6.1.36 we have
Let \(r_k=\mathrm{rank\, im}\,\partial _k\). From (6.45) and the exactness of (6.44), we have
Evidently, \(Q(t)={\underset{\mathrm {k\geqslant 0}}{\sum }}r_kt^k, t\in \mathbb R\), is the desired polynomial. \(\square \)
The next theorem establishes the so-called “Morse relation”.
Theorem 6.2.20
If X is a Banach space, \(\varphi \in C^1(X)\), \(a, b\in \mathbb R\backslash \varphi (\{K_{\varphi })\), \(a<b\), \(\varphi ^{-1}((a, b))\) contains a finite number of critical points \(\{u_i\}^n_{i=1}\) and \(\varphi \) satisfies the \(C_c\)-condition for every \(c\in \left[ a, b\right) \), then
-
(a)
for all \(k\in \mathbb N_0\), we have \(M_k(a,b)\geqslant \beta _k(a, b)\);
-
(b)
if the Morse-type numbers \(M_k(a, b)\) are finite for all \(k\in \mathbb N_0\) and vanish for all large \(k\in \mathbb N_0\), then so do the Betti numbers \(\beta _k(a, b)\) and we have
$${\underset{\mathrm {k\geqslant 0}}{\sum }}M_k(a, b)t^k={\underset{\mathrm {k\geqslant 0}}{\sum }}\beta _k(a, b)t^k+(1+t)Q(t)\ \text {for all}\ t\in \mathbb R,$$where Q(t) is a polynomial in \(t\in \mathbb R\) with nonnegative integer coefficients.
Proof
(a) Let \(c_k=\varphi (u_k)\) for all \(k\in \{1,\ldots , n\}\) and pick \(\{\vartheta _k\}^n_{k=0}\subseteq [a, b]\backslash \varphi (K_{\varphi })\) such that
Then from Definition 6.2.18 and Propositions 6.2.16 and 6.2.17, we have
(b) If \(M_k(a, b)\) is finite for all \(k\in \mathbb N_0\) and vanishes for large \(k\in \mathbb N_0\), then from (6.46) it is clear that so do the Betti numbers \(\beta _k(a, b),\beta _k(\vartheta _{i-1},\vartheta _i)\). Then using Lemma 6.2.19 we have
where Q(t) is a polynomial in \(t\in \mathbb R\) with nonnegative integer coefficients. From (6.46) and (6.47) we conclude that
The proof is now complete.\(\square \)
Remark 6.2.21
If in (6.48) we choose \(t=-1\), then
and this equality is known as the “Poincaré–Hopf formula”.
When the functional \(\varphi \in C^1(X)\) has critical values which are bounded from below and satisfy the C-condition, then the global behavior of \(\varphi \) can be described by the critical groups of \(\varphi \) at infinity.
Definition 6.2.22
Let \(\varphi \in C^1(X)\) and assume that \(\varphi \) satisfies the C-condition and \(\inf \varphi (K_{\varphi })>-\infty \). The “critical groups of \(\varphi \) at infinity” are defined by
with \(c<\inf \varphi (K_{\varphi })\).
Remark 6.2.23
Corollary 5.3.13 reveals that the above definition is independent of the choice of the level \(c<\inf \varphi (K_{\varphi })\). Indeed, if \(d<c<\inf \varphi (K_{\varphi })\), then from Corollary 5.3.13 we know that \(\varphi ^d\) is a strong deformation retract of \(\varphi ^c\). So, Corollary 6.1.24 (a) implies that \(H_k(X,\varphi ^c)=H_k(X,\varphi ^d)\) for all \(k\in \mathbb N_0\).
Proposition 6.2.24
If X is a Banach space, \(\varphi \in C^1(X)\) and \(\varphi \) satisfies the C-condition then
-
(a)
for \(\varphi (\cdot )\) bounded from below, we have
$$C_k(\varphi ,\infty )=\delta _{k, 0}\mathbb Z\ \text {for all}\ k\in \mathbb N_0;$$ -
(b)
for \(\varphi (\cdot )\) unbounded from below and \(\inf \varphi (K_{\varphi })>-\infty \), we have \(C_k(\varphi ,\infty )=\tilde{H}_{k-1}(\varphi ^c)\) for all \(k\in \mathbb N_0\) and all \(c<\inf \varphi (K_{\varphi })\).
Proof
(a) Let \(c<\inf \varphi (X)\). Then \(\varphi ^c=\emptyset \) and so by Definition 6.2.22 we have
(see Definition 6.1.12, Axiom 7 and Remark 6.1.13).
(b) From Definition 6.2.22 and since X is contractible, we see that the reduced homology groups of X are trivial for all \(k\in \mathbb N_0\). We consider the following long exact sequence
From the exactness of (6.49) and since \(\tilde{H}_k(X)=\tilde{H}_{k-1}(X)\) for all \(k\in \mathbb N_0\), we infer that
The proof is now complete.\(\square \)
Remark 6.2.25
In particular, in the setting of part (b), we have \(C_0(\varphi ,\infty )=0\).
Proposition 6.2.26
If X is a Banach space, \(\varphi \in C^1(X),\ \varphi \) satisfies the C-condition and \(K_{\varphi }\) is finite, then \(\mathrm{rank}\,C_k(\varphi ,\infty )\leqslant {\underset{\mathrm {u\in K_{\varphi }}}{\sum }}\mathrm{rank}\,C_k(\varphi , u)\) for all \(k\in \mathbb N_0\).
Proof
This proposition is an immediate consequence of Proposition 6.2.17 with \(b>\sup \varphi (K_{\varphi })\) and \(a<\inf \varphi (K_{\varphi })\). \(\square \)
Corollary 6.2.27
If X is a Banach space, \(\varphi \in C^1(X),\ \varphi \) satisfies the C-condition, \(K_{\varphi }\) is finite and \(C_{k_0}(\varphi ,\infty )\ne 0\) for some \(k_0\in \mathbb N_0\), then there exists a \(u\in K_{\varphi }\) such that \(C_{k_0}(\varphi , u)\ne 0\).
Proposition 6.2.28
If X is a Banach space, \(\varphi \in C^1(X)\) and satisfies the C-condition, \(\inf \varphi (K_\varphi )>-\infty \), then
-
(a)
for \(-\infty<a<\inf \varphi (K_{\varphi })\leqslant \sup \varphi (K_{\varphi })<b\), we have
$$C_k(\varphi ,\infty )=H_k(\varphi ^b,\varphi ^a)\ \text {for all}\ k\in \mathbb N_0;$$ -
(b)
\(C_k(\varphi ,\infty )=0\) for all \(k\in \mathbb N_0\) when \(K_{\varphi }=\emptyset \);
-
(c)
\(C_k(\varphi ,\infty )=C_k(\varphi , u)\) for all \(k\in \mathbb N_0\) when \(K_{\varphi }=\{u\}\).
Proof
-
(a)
This is a consequence of Proposition 6.2.16.
-
(b)
Follows from part (a) with \(a=b\) (see Corollary 6.1.16).
-
(c)
Follows from part (a) and Proposition 6.2.16.
\(\square \)
Also, from Theorem 6.2.20(b) and Proposition 6.2.28(a), we infer a global version of the Morse relation.
Theorem 6.2.29
If X is a Banach space, \(\varphi \in C^1(X),\ \varphi \) satisfies the C-condition, \(K_{\varphi }\) is finite, \(C_k(\varphi , u)\) has a finite rank for all \(k\in \mathbb N_0\) and all \(u\in K_{\varphi }\), and vanishes for large \(k\in \mathbb N_0\), then
with Q(t) a polynomial with nonnegative integer coefficients.
Next, we discuss the critical groups at infinity in more detail. First we consider functionals which exhibit some kind of local linking at infinity.
Proposition 6.2.30
If X is a Banach space with \(X=Y\oplus V\) with \(\mathrm{dim}\, Y<\infty ,\ \varphi \in C^1(X),\ \varphi \) satisfies the C-condition, \(\inf \varphi (K_{\varphi })>-\infty ,\ \varphi |_{V}\) is bounded from below and \(\varphi |_Y\) is anticoercive (that is, if \(y\in Y,\ ||y||\rightarrow \infty \), then \(\varphi (y)\rightarrow -\infty \)), then \(C_d(\varphi ,\infty )\ne 0\) with \(d=\mathrm{dim}\, Y\).
Proof
Let \(c<\min \{\inf \varphi |_V,\inf \varphi (K_{\varphi })\}\). Since by hypothesis \(\varphi |_{Y}\) is anticoercive, we can find large \(r>0\) such that
So, we have
Consider the deformation \(h:[0,1]\times (X\setminus V)\rightarrow X\setminus V\) defined by
It follows that \(\partial B^Y_{\rho }\) is a strong deformation retract of \(X\setminus V\). Hence
(see Proposition 6.1.15). We consider the following commutative diagram
with \(i_*, j_*,\vartheta _*,\eta _*\) being the group homomorphisms induced by the corresponding inclusion maps. In (6.51) the top row is exact (see Proposition 6.1.23). We have \(i_*=\eta _*\circ \vartheta _*\) and from (6.50) we see that \(i_*=0\). The exactness of the top row implies that \(j_*\) is injective for all \(k\in \mathbb N_0\). From the reduced exact homology sequence (see Proposition 6.1.29) we have
Since Y is finite-dimensional, we have
The proof is now complete.\(\square \)
The next two results provide some further information about critical groups at infinity in the context of Hilbert spaces. The proofs of these results can be found in Bartsch and Li [38].
So, let H be a Hilbert space and \(\varphi \in C^1(H)\). We introduce the following condition on \(\varphi \).
\((A_{\infty })\) \(\varphi (u)=\frac{1}{2}(A(u), u)_H+\psi (u)\) for all \(u\in H\), with \(A\in \mathscr {L}(H, H)\) self adjoint, 0 is isolated in the spectrum of A, \(\psi \in C^1(H),\ \lim \limits _{||u||\rightarrow \infty }\frac{\psi (u)}{||u||^2}=0\) (subquadratic), \(\psi \) and \(\psi '\) are bounded (that is, map bounded sets to bounded sets) and \(\varphi \) is bounded from below and satisfies the C-condition.
Remark 6.2.31
If \(A_{\infty }\) holds, then we set
The space V admits an orthogonal direct sum decomposition
with \(V_+, V_-\) being A-invariant, \(A|_{V_-}<0\) and \(A|_{V_+}>0\). So, we can find \(c_0>0\) such that
Let \(m=\mathrm{dim}\, V_-\) [the Morse index of \(\varphi \) at infinity, compare with Proposition 3.4.18(b)] and \(\nu =\mathrm{dim}\, Y\) (known as the nullity of \(\varphi \) at infinity).
The results of Bartsch–Li [38] mentioned earlier read as follows.
Theorem 6.2.32
If H is a Hilbert space and \(\varphi \in C^1(H)\) satisfies condition \((A_{\infty })\) above, then \(C_k(\varphi ,\infty )=0\) for all \(k\notin \{m, m+1,\ldots , m+\nu \}\).
Remark 6.2.33
In this theorem we do not require that \(m,\nu \) are finite. If \(m\in \mathbb N\) and \(\nu =0\), then \(C_k(\varphi ,\infty )=\delta _{k, m}\mathbb Z\) for all \(k\in \mathbb N_0\).
Imposing the so-called angle conditions on \(\varphi \), we derive more information concerning the critical groups at infinity.
Theorem 6.2.34
If H is a Hilbert space, \(\varphi \in C^1(H)\), \(\varphi \) satisfies \((A_{\infty })\) and \(m,\nu \in \mathbb N\), then
-
(a)
\(C_k(\varphi ,\infty )=\delta _{k, m}\mathbb Z\) for all \(k\in \mathbb N_0\), when \(\varphi \) satisfies the following “angle condition”:
$$(A^+_{\infty }) ``{\text {there exist }} M>0 {\text { and }}\vartheta \in (0,1){\text { such that}}$$$$\left\langle \varphi '(u), y\right\rangle \geqslant 0\ \text {for all}\ u=y+v\in H,\ y\in Y, v\in V$$$${\text {with }}||u||\geqslant M{\text { and }}||v||\leqslant \vartheta ||u||''.$$ -
(b)
\(C_k(\varphi ,\infty )=\delta _{k, m+\nu }\mathbb Z\) for all \(k\in \mathbb N_0\), when \(\varphi \) satisfies the following angle condition
$$(A^-_{\infty }) ``{\text {there exist }}M>0{\text { and }}\vartheta \in (0,1){\text { such that}}$$$$-\left\langle \varphi '(u), y\right\rangle \geqslant 0\ \text {for all}\ u=y+v\in H,y\in Y, v\in V$$$${\text {with }}||u||\geqslant M{\text { and }}||v||\leqslant a||u||.''$$
We will derive similar information for the critical groups at an isolated critical point. To this end, we prove three auxiliary results.
Lemma 6.2.35
If X is a reflexive Banach space, \(\varphi \in C^1(X),\ \varphi \) satisfies the C-condition and \(u_0\in K_{\varphi }\) is isolated with \(c=\varphi (u_0)\) isolated in \(\varphi (K_{\varphi })\), then there exist \(\psi \in C^1(X)\), \(U\subseteq X\) open with \(u_0\in U\) and \(\delta >0\) such that
-
(a)
\(\psi \) satisfies the C-condition;
-
(b)
\(\varphi \leqslant \psi \) and \(\varphi |_{U}=\psi |_{U};\)
-
(c)
\(K_{\varphi }=K_{\psi }\);
-
(d)
\(K_{\psi }\cap \psi ^{-1}([c-\delta , c+\delta ])=\{u_0\};\)
-
(e)
if \(X=H=\) a Hilbert space and \(\varphi \in C^{\rho }(H)\) with \(\rho \geqslant 2\), then we have \(\psi \in C^{\rho }(H)\) too.
Proof
Thanks to the Troyanski renorming theorem (see Theorem 2.7.36), we may assume that X and \(X^*\) are locally uniformly convex with Fréchet differentiable norms (except at the origin). Then the map \(h:X\rightarrow \mathbb R_+\) defined by
is of class \(C^1\) and we have
with \(J:X\rightarrow X^*\) being the duality map (see Definition 2.7.21 and Proposition 2.7.33). Then given \(0<\rho _1<\rho _2\) such that \(\bar{B}_{\rho _2}(u_0)\cap K_{\varphi }=\{u_0\}\) and \(\varphi ,\varphi '\) restricted to \(\bar{B}_{\rho _2}(u_0)\) are bounded, we can find \(\eta \in C^1(X)\) such that
The existence of such a function is easily seen if we recall that the smoothness of X implies the existence of a \(C^1\)-bump function (recall that a bump function on X is a function on X with nonempty bounded support).
Let \(U=B_{\rho _1}(u_0)\). Because \(\varphi \) satisfies the C-condition, we can find \(\gamma >0\) such that
Recall that \(c=\varphi (u_0)\) is isolated in the critical values \(\varphi (K_{\varphi })\). Then we can find \(c_0\in \left( c-\frac{\gamma }{2M}, c\right) \) and \(\delta >0\) such that
(that is, \([c_0-\delta , c_0+\delta ] \) is a regular interval). We set
Evidently, \(\psi \in C^1(X)\). We claim that \((\psi , U,\delta )\) as above is the desired triple postulated by the lemma
(a) Suppose that \(\{u_n\}_{n\geqslant 1}\subseteq X\) is a sequence such that
We have
From (6.56) and (6.57), we see that we can find \(n_0\in \mathbb N\) such that
From (6.55) we see that \(\varphi \leqslant \psi \). So, because of (6.56) we infer that
From (6.56), (6.58), (6.59) and since \(\varphi \) satisfies the C-condition, we conclude that \(\{u_n\}_{n\geqslant 1}\subseteq X\) admits a strongly convergent subsequence. Therefore \(\psi \) satisfies the C-condition.
(b) This follows at once from (6.53) and (6.55).
(c) Recall that
[see (6.53), (6.55) and (6.57)]. Therefore we see that
(d) Let \(u\in K_{\varphi }\backslash \{u_0\}\). Then from part (c) we have \(u\in K_{\varphi }\). Recalling the choice of \(\rho _2\), we see that \(||u||>\rho _2\). Because of (6.53) we have
From the choices of \(c_0\) and \(\delta \) it follows that
(e) Since \(h(u)=\frac{1}{2}||u||^2\) for all \(u\in X=H\) is \(C^{\infty }\), we have \(\eta \in C^{\infty }(H)\). So, if \(\varphi \in C^{\rho }(H)\), \(\rho \geqslant 2\), then \(\psi \in C^{\rho }(H)\) [see (6.55)]. \(\square \)
Remark 6.2.36
For every \(\vartheta >0\), the set \((c-\vartheta , c)\backslash \varphi (K_{\varphi })\) is open. So, in Lemma 6.2.35 we can replace the hypothesis that \(c=\varphi (u_0)\) is isolated in \(\varphi (K_{\varphi })\) by a weaker one which says that there is a sequence \(\{c_n\}_{n\geqslant 1}\subseteq \mathbb R\backslash \varphi (K_{\varphi })\) with \(c_n<c\) and \(c_n\rightarrow c\).
Lemma 6.2.37
If \(\varphi \in C^2(\mathbb R^N),\ U\subseteq \mathbb R^N\) is open and bounded and \(K\subseteq U\) is compact such that \(K_{\varphi }\cap \overline{(U\backslash K)}=\emptyset \), then for every \(\epsilon >0\), we can find \(\psi \in C^2(\mathbb R^N)\) such that
-
(a)
\(|\varphi (u)-\psi (u)|+||\varphi '(u)-\psi '(u)||_*\leqslant \epsilon \) for all \(u\in \mathbb R^N\);
-
(b)
\(\varphi |_{\mathbb R^N\backslash U}=\psi |_{\mathbb R^N\backslash U}\);
-
(c)
\(K_{\psi }\cap \bar{U}\) is finite and all its elements are nondegenerate critical points.
Proof
By hypothesis, we have
Choose \(\eta \in C^{\infty }(\mathbb R^N)\) such that
Also, let \(\rho >0\) and \(\vartheta >0\) such that
By Sard’s theorem (see Theorem 3.1.16), we can find \(e\in \mathbb R^N\) such that
(that is, \(\varphi ''(u)\) is nondegenerate whenever \(\varphi '(u)=-e\)). We consider \(\psi \in C^2(\mathbb R^N)\) defined by
We have
From the choice of \(\eta \) and e, we see that \(\psi \) defined above satisfies statements (a) and (b) of the lemma. Also, we have
Let \(u\in K_{\psi }\cap \bar{U}\). Then from (6.62) it follows that \(u\in \mathrm{int}\, K\). Hence from (6.61) it follows that
Since e is not a critical value of \(\varphi '\), we infer that \(\varphi ''(u)=\psi ''(u)\) is invertible. Therefore the elements of \(K_{\psi }\cap \bar{U}\) are nondegenerate, hence isolated (by the inverse function theorem) and located in the compact set K. Therefore \(K_{\psi }\cap \bar{U}\) is also finite. This proves part (c). \(\square \)
Lemma 6.2.38
If \(\varphi \in C^2(\mathbb R^N)\), \(u_0\in K_{\varphi }\) is isolated and \(c=\varphi (u_0)\), then we can find \(\psi \in C^2(\mathbb R^N)\) such that
-
(a)
\(\varphi =\psi \) is a neighborhood of \(u_0\);
-
(b)
\(K_{\psi }\) is finite;
-
(c)
\(K^c_{\psi }=\{u_0\}\);
-
(d)
\(\psi \) is coercive (hence it satisfies the C-condition).
Proof
By modifying \(\varphi \) outside a ball centered at \(u_0\), if necessary, we may assume that \(\varphi \) is coercive and there exists a \(\rho >0\) such that \(K_{\varphi }\cap (\mathbb R^N\backslash B_{\rho }(u_0))=\emptyset \). Let \(r\in (0,\rho )\) be such that \(K_{\varphi }\cap \overline{B_r(u_0)}=\{u_0\}\). Let \(\hat{\psi }\in C^2(\mathbb R^N)\) be the function obtained in Lemma 6.2.37, with \(U=B_{2\rho }(u_0)\backslash \overline{B_{\frac{r}{2}}(u_0)}\), \(K=\bar{B}_{\rho }(u_0)\backslash B_r(u_0)\) and any \(\epsilon >0\). Evidently, \(\hat{\psi }\) satisfies parts (a),(b),(c) of the lemma. Finally apply Lemma 6.2.35 to \(\hat{\psi }\) and denote by \(\psi _0\) the function we obtain in this way. Then \(\psi _0\) satisfies (a)–(d) in the lemma. \(\square \)
We will use these lemmata to derive some useful consequences concerning the critical groups of isolated critical points for \(C^2\)-functions.
Proposition 6.2.39
If \(\varphi \in C^2(\mathbb R^N)\) and \(u_0\in K_{\varphi }\) is isolated, then \(\mathrm{rank}\,C_k(\varphi , u_0)<\infty \) for all \(k\in \mathbb N_0\) and \(C_k(\varphi , u_0)=0\) for all \(k\notin \{0,1\ldots , N\}\).
Proof
From Definition 6.2.1 and Remark 6.2.2, we know that the critical groups \(C_k(\varphi , u_0)\), \(k\in \mathbb N_0\), depend only the local structure of \(\varphi \). So, using Lemma 6.2.38 we see that without any loss of generality, we may assume that \(\varphi \) is coercive (hence it satisfies the C-condition), \(K_{\varphi }\) is finite and \(K^{c_0}_{\varphi }=\{u_0\}\), where \(c_0=\varphi (u_0)\). Let \(a, b\in \mathbb R\) such that \(a<c_0=\varphi (u_0)<b\) and \(K_{\varphi }\cap \varphi ^{-1}([a, b])=\{u_0\}\). Invoking Proposition 6.2.16, we have
Let \(r>0\) be such that
Let \(U=B_r(u_0)\) and \(K=\overline{B_{r/2}(u_0)}\). Then
Pick \(\epsilon >0\) such that
and let \(\psi \in C^2(\mathbb R^N)\) be as postulated by Lemma 6.2.37 for the aforementioned choices of U, K and \(\epsilon >0\). Then from Lemma 6.2.37, we have
Then from (6.63), (6.64) and Theorem 6.2.20 (the Morse relation), we have
with \(\{u_i\}^n_{i=1}=K_{\psi }\cap U\). Each \(u_i\) is a nondegenerate critical point for \(\psi \) and so from Proposition 6.2.6, we obtain
This proves the proposition. \(\square \)
As a consequence of Proposition 6.2.39 and of Theorem 6.2.13 (the shifting theorem), we have:
Corollary 6.2.40
If H is a Hilbert space, \(\varphi \in C^2(H)\) and \(u_0\in K_{\varphi }\) is isolated with finite Morse index m and \(\nu =\mathrm{dim ker}\,\varphi ''(u_0)<+\infty \), then \(\mathrm{rank}\,C_k(\varphi , u_0)\) is finite for all \(k\in \mathbb N_0\) and \(C_k(\varphi , u_0)=0\) for all \(k\notin \{m,\ldots , m+\nu \}\).
The next proposition is useful in obtaining nontrivial critical points with a nontrivial critical group.
Proposition 6.2.41
If X is a Banach space, \(\varphi \in C^1(X),\ \varphi \) satisfies the C-condition, \(K_{\varphi }\) is finite with \(0\in K_{\varphi }\) and for some \(k\in \mathbb N_0\) we have
then there exists a \(u\in K_{\varphi }\backslash \{0\}\) such that \(C_k(\varphi , u)\ne 0\).
Proof
From Corollary 6.2.27 we know that there exists a \(u\in K_{\varphi }\) such that
On the other hand, by hypothesis,
Comparing (6.65) and (6.66), we conclude that \(u\ne 0\). \(\square \)
Proposition 6.2.42
If X is a Banach space, \(\varphi \in C^{1}(X)\), \(\varphi \) satisfies the C-condition, \(K_{\varphi }\) is finite with \(0\in K_{\varphi }\) and for some \(k\in \mathbb N_0\) we have
then there exists a \(u\in K_{\varphi }\) such that
Proof
Without any loss of generality we may assume that \(\varphi (0)=0\). Choose \(\epsilon >0\) small such that \(\varphi (K_{\varphi })\cap [-\epsilon ,\epsilon ]=\{0\}\). Let \(c<\min \left\{ -\epsilon , \inf \varphi (K_{\varphi })\right\} \). From Proposition 6.2.17 we have
while from Definition 6.2.22 and the choice of c we have
We consider the sets \(\varphi ^c\subseteq \varphi ^{-\epsilon }\subseteq \varphi ^{\epsilon }\subseteq X\) and use Proposition 6.1.37. Then
In the first case, by Proposition 6.2.15 we can find \(u\in K_{\varphi }\) such that
Similarly, in the second case, we can find \(\tilde{u}\in K_{\varphi }\) such that
The proof is now complete.\(\square \)
We conclude this section by relating critical groups with the Leray–Schauder degree. We start with the definition of the Leray–Schauder index.
Definition 6.2.43
Let X be a Banach space, \(\varphi =i-f:X\rightarrow X\) with f compact and \(u_0\in X\) be an isolated solution of the equation \(\varphi (u)=0\). Let \(r>0\) be such that \(u_0\) is the only solution of the equation in \(\bar{B}_r(u_0)\). The “Leray–Schauder index of \(\varphi \) at \(u_0\)” is defined by
Suppose that \(X=H=\) a Hilbert space, \(\varphi \in C^1(H)\) and \(\nabla \varphi =i-f\) with \(f:H\rightarrow H\) compact (here by \(\nabla \varphi (\cdot )\) we denote the gradient of \(\varphi \)). Note that both \(C_k(\varphi , u_0)\) and \(i_{LS}(\nabla \varphi , u_0)\) are topological invariants describing the local behavior at an isolated critical point \(u_0\in K_{\varphi }\). So, it is reasonable to expect that the two quantities are related. The precise relation is given in the next proposition, the proof of which can be found in Chang [119] (Theorem 3.2, p. 100).
Proposition 6.2.44
If H is a Hilbert space, \(\varphi \in C^2(H)\), \(\varphi \) satisfies the C-condition, \(\nabla \varphi =i-f\) with \(f:H\rightarrow H\) a compact map and \(u_0\in K_{\varphi }\) is isolated, then \(i_{LS}(\nabla \varphi , u_0)={\underset{\mathrm {k\in \mathbb N_0}}{\sum }}(-1)^k\mathrm{rank}\,C_k(\varphi , u_0)\).
Remark 6.2.45
This proposition reveals that for potential compact vector fields, the critical groups provide more information than the Leray–Schauder index.
6.3 Continuity and Homotopy Invariance of Critical Groups
In this section we show that critical groups are continuous with respect to the \(C^1\)-topology and are invariant under homotopies which preserve the isolation of the critical point.
We start with a definition.
Definition 6.3.1
Let X be a Banach space, \(\varphi \in C^1(X),\ C\subseteq X\) a nonempty closed subset. We say that \(\varphi \) satisfies the “PS-condition over C” if every sequence \(\{u_n\}_{n\geqslant 1}\subseteq C\) such that \(\{\varphi (u_n)\}_{n\geqslant 1}\) is bounded and \(\varphi '(u_n)\rightarrow 0\) in \(X^*\) as \(n\rightarrow \infty \) admits a strongly convergent subsequence.
The next lemma is a property of the negative pseudogradient flow. So, let \(V:X\backslash K_{\varphi }\rightarrow X\) be a pseudogradient vector field corresponding to \(\varphi \in C^1(X)\) (see Theorem 5.1.4). We consider the abstract Cauchy problem
Let \(\left[ 0,\eta _+(x)\right) \) be the maximal interval of existence for (6.69).
Lemma 6.3.2
If X is a Banach space, \(\varphi \in C^1(X),\ u\in K_{\varphi }\) is isolated and \(\varphi \) satisfies the PS-condition over a closed neighborhood C of u, then there exists \(\epsilon >0\) and a neighborhood D of u such that if \(x\in D\), then either \(\sigma (t, x)\in C\) for all \(t\in (0,\eta _+(x))\) or \(\sigma (t, x)\in C\) until \(\varphi (\sigma (t, x))\) becomes less than \(\varphi (u)-\epsilon \).
Proof
Let \(r>0\) be such \(\bar{B}_r(u)\subseteq C,\ \varphi |_{\bar{B}_r(u)}\) is bounded and if \(A=\{v\in X:r/2\leqslant ||v-u||\leqslant r\}\), then \(A\cap K_{\varphi }=\emptyset \). The PS-condition and the definition of the pseudogradient vector field (see Definition 5.1.1) imply that
We set \(D=\bar{B}_{r/2}(u)\cap \varphi ^{c+\vartheta \frac{r}{4}}\), where \(c=\varphi (u)\). Let \(x\in D\) be such that \(\sigma (t, x)\) does not stay in C for all \(t\in (0,\eta _+(x))\). So, there exist \(0\leqslant t_1< t_2<\eta _+(x)\) such that
Then we have
We finish the proof by taking \(\epsilon =\frac{\vartheta r}{4}\). \(\square \)
This lemma leads to some other useful observations concerning the pseudogradient flow.
Lemma 6.3.3
If X is a Banach space, \(\varphi \in C^1(X),\ u\in K_{\varphi }\) is isolated and \(\varphi \) satisfies the PS-condition over a closed ball \(\bar{B}_r(u)\), then there exist \(\epsilon >0\) and \(E\subseteq X\) such that
-
(a)
E is a closed neighborhood of u;
-
(b)
E is positively invariant for the pseudogradient flow \(\sigma (\cdot )\);
-
(c)
\(\varphi ^{-1}([c-\epsilon , c+\epsilon ])\cap E\) is complete, where \(c=\varphi (u)\);
-
(d)
the PS-condition is satisfied over \(\varphi ^{-1}([c-\epsilon , c+\epsilon ])\cap E\).
Proof
Let \(C=\bar{B}_r(u)\) and let \(\epsilon >0\) and D, a neighborhood of u, be as postulated by Lemma 6.3.2. Consider the set
where \(\sigma (\cdot , v)\) is the pseudogradient flow emanating from \(v\in D\) and \([0,\eta _+(v))\) is the maximal interval of existence of the flow. We set
Evidently, E is a closed neighborhood of u which is positively invariant for the flow. So, we have proved (a) and (b). From Lemma 6.3.2 we have
So, \(\varphi ^{-1}([c-\epsilon , c+\epsilon ])\cap E\) is complete (being closed). This proves (c) and because by hypothesis \(\varphi |_{\bar{B}_r(u)}\) satisfies the PS-condition (see Definition 6.3.1), we conclude that (d) holds. \(\square \)
In what follows, for \(\varphi \in C^1(X), c\in \mathbb R\) and \(A\subseteq X\), we set
Theorem 6.3.4
If X is a separable reflexive Banach space, \(\varphi ,\psi \in C^1(X),\ u\in X\), there exists an \(r>0\) such that \(\bar{B}_r(u)\cap K_{\varphi }=\bar{B}_r(u)\cap K_{\psi }=\{u\}\) and both \(\varphi \) and \(\psi \) satisfy the PS-condition on \(\bar{B}_r(u)\), then there exists a \(\delta >0\) depending only on \(\varphi \) such that
Proof
Let \(\epsilon >0\) and \(E\subseteq X\) be as postulated by Lemma 6.3.3. From Definition 6.2.18 and Theorem 6.2.20, we have
Let \(\rho >0\) be such that
Since by hypothesis \(\varphi \) satisfies the PS-condition on \(\bar{B}_r(u)\), we have
Choose \(h\in C^1(X)\) such that
This is a smooth bump function which exists since X is a separable reflexive Banach space. Let \(\delta =\min \left\{ \frac{\epsilon }{3},\frac{n}{2(1+\eta )}\right\} \). We introduce the function \(\hat{\psi }\in C^1(X)\) defined by
Since \(||\varphi -\psi ||_{C^1(X)}\leqslant \delta \), we have
Also, from (6.75) we have
From (6.75) and (6.76) we see that
Therefore \(\hat{\psi }^{-1}([c-\epsilon , c+\epsilon ])\cap E=\varphi ^{-1}([c-\epsilon , c+\epsilon ])\cap E\) is complete. From (6.77) it is clear that \(\hat{\psi }\) satisfies the PS-condition over \(\hat{\psi }^{-1}([c-\epsilon , c+\epsilon ])\cap E\). Moreover, \(\bar{B}_{\rho }(u)\subseteq \mathrm{int}\, E\), so that E is positively invariant for the pseudogradient flow \(\hat{\sigma }(t)\) corresponding to the functional \(\hat{\psi }\in C^1(X)\). From (6.76) it is clear that \(K_{\hat{\psi }}=\{u\}\). Then we have
But from (6.75), (6.76) and the local character of the critical groups, we have
Then from (6.72), (6.79), (6.80) we conclude that
The proof is now complete.\(\square \)
Remark 6.3.5
If we choose a field \(\mathbb {F}\) for the coefficients of the homology groups in the definition of the critical groups of \(\varphi \) (see Definition 6.2.1), then we know that the critical groups are in fact \(\mathbb {F}\)-vector spaces (for example, we can take \(\mathbb {F}=\mathbb R\), see Remark 6.2.2). Hence we avoid torsion phenomena and in the above theorem, we can say that \(C_k(\varphi ,u)=C_k(\psi , u)\) for all \(k\in \mathbb N_0\).
As a consequence of Theorem 6.3.4, we see that the critical groups are invariant under homotopies which preserve the isolation of the critical point.
Theorem 6.3.6
If X is a separable reflexive Banach space, \(\{h_t\}_{t\in [0,1]}\subseteq C^1(X), u\in X\) there exists \(r>0\) such that \(\bar{B}_r(u)\cap K_{h_t}=\{u\}\) for all \(t\in [0,1]\), all \(\{h_t\}_{t\in [0,1]}\) satisfy the PS-condition on \(\bar{B}_r(u)\) and \(t\rightarrow h_t\) is continuous from [0, 1] into \(C^1(X)\), then for all \(k\in \mathbb N_0\ C_k(h_t, u)\) is independent of \(t\in [0,1]\).
Proof
This follows at once from Theorem 6.3.4 and the compactness of [0, 1]. \(\square \)
Remark 6.3.7
Under the hypotheses of Theorem 6.3.6, if u is a local minimizer of \(h_0\), then it is also a local minimizer for all \(h_t\), \(t\in \left( 0,1\right] \). In fact the above result remains valid if X is only a Banach space (see Theorem 6.3.8 below).
Theorem 6.3.8
If X is a Banach space, \(\{h_t\}_{t\in [0,1]}\subseteq C^1(X)\), each \(h_t\) satisfies the PS-condition, \(a, b:[0,1]\rightarrow \mathbb R\) are continuous functions such that \(a(t)<b(t)\) for all \(t\in [0,1]\), both a(t), b(t) are regular values of \(h_t\), \(t\in [0,1]\), and \(t\rightarrow h_t\) is continuous from [0, 1] into \(C^1(X)\), then for all \(k\in \mathbb N_0\), \(H_k\left( h^{b(t)}_{t}, h^{a(t)}_{t}\right) \) is independent of \(t\in [0,1]\).
Proof
To simplify an already cumbersome notation, for every \(t_0,t_1\in [0,1]\) instead of
we write
Suppose that \(|t_1-t_0|\) is small. Since by hypothesis \(h_0\) satisfies the PS-condition, we can find \(c<d\)
The continuity of \(h_0\) implies that we can find \(\delta >0\) such that
where \((K_0)_{\delta }=\{u\in X:d(u, K_0)<\delta \}\) (the \(\delta \)-neighborhood of the set \(K_0\)). The PS-condition implies that there exists an \(\epsilon =\epsilon (\delta )>0\) such that
Since \(|t_1-t_0|\) is small we have
We can construct a pseudogradient vector field for h, which coincides with a pseudogradient vector field for \(h_0\) on \(\left( h^{b_1}_{1}\cap h^{b_0}_{0}\right) \backslash (K_0)_{\delta }\). Then according to Corollary 5.3.13, we have that
In a similar fashion, we also show that
So, using Proposition 6.1.18, we have
On the other hand, again from Corollary 5.3.13 we have
From (6.81), (6.82), (6.83) and Corollary 6.1.24, we infer that
Finally the conclusion of the theorem follows from the compactness of [0, 1]. \(\square \)
Remark 6.3.9
A careful reading of this proof reveals that we may have \(b(t)=+\infty \) for all \(t\in [0,1]\) (hence \(h^{b(t)}_t=X\) for all \(t\in [0,1]\)).
6.4 Extended Gromoll–Meyer Theory
In this section we present the Gromoll–Meyer theory of dynamically isolated critical sets, which is useful in dealing with resonant elliptic problems.
Definition 6.4.1
Let (X, d) be a metric space. A “flow” on X is a continuous map \(\sigma :\mathbb R\times X\rightarrow X\) such that
-
(a)
\(\sigma (0,u)=u\) for all \(u\in X\);
-
(b)
\(\sigma (t_1,\sigma (t_2,u))=\sigma (t_1+t_2,u)\) for all \(t_1,t_2\in \mathbb R\), all \(u\in X\) (the group property).
Remark 6.4.2
It is an easy consequence of the group property that, for each \(t\in \mathbb R\), \(\sigma (t,\cdot )\) is a homeomorphism of X onto X (that is, a bicontinuous bijection).
Given a flow \(\sigma (t, u)\), a set \(C\subseteq X\) and \(b>0\), we introduce the following sets:
Also, given \(t_1\leqslant 0\leqslant t_2\) and \(0\leqslant t_0\leqslant +\infty \), we define
From these definitions, we easily deduce the following lemma.
Lemma 6.4.3
(a) \(G^t(C)=G^t(\bar{C})\) for all \(t\geqslant 0\).
(b) \(G^{t_1}(C)\subseteq G^{t_2}(C)\) if \(t_2\leqslant t_1\).
(c) \(G^t(C_1)\subseteq G^t(C_2)\) for all \(t\geqslant 0\) if \(C_1\subseteq C_2\).
(d) \(G^t(C)\) is closed in X for every \(t\geqslant 0\).
(e) \(G^{t_1+t_2}(C)=G^{t_2}(G^{t_1}(C))\) for all \(t_1,t_2\geqslant 0\).
(f) If \(G^t(C)\subseteq \mathrm{int}\, C\), then \(G^{2t}(C)\subseteq \mathrm{int}\, G^t(C)\).
Proof
Only part (f) is not obvious. Suppose that the implication is not true. So, we have \(G^{2t}(C)\cap \partial (G^t(C))\ne \emptyset \). Let \(\hat{u}\in G^{2t}(C)\cap \partial (G^t(C))\). So,
Hence we can find a sequence \(\{t_n\}_{n\geqslant 1}\subseteq [-t, t]\) such that
By passing to a subsequence if necessary, we may assume that \(t_n\rightarrow \hat{t}\in [-t, t]\). Then \(\sigma (\hat{t},\hat{u})\in \partial C\). But \(\hat{u}\in G^{2t}(C)\) and so \(\sigma ([-2t, 2t],\hat{u})\subseteq \bar{C}\), which implies that \(\sigma (\hat{t},\hat{u})\in G^t(\bar{C})=G^t(C)\subseteq \mathrm{int}\, C\), a contradiction. \(\square \)
We introduce the following family of closed sets
Definition 6.4.4
Let (X, d) be a metric space and \(\sigma (t, u)\) a flow on it.
-
(a)
A set \(C\subseteq X\) is said to be “\(\sigma \)-invariant” if for all \(u\in C\) and for all \(t\in \mathbb R\), we have \(\sigma (t, u)\in C\).
-
(b)
A \(\sigma \)-invariant set C is said to be “isolated” if there is a neighborhood U of C such that
$$U\in \Sigma \ \text {and}\ I(U)=C.$$
In this case, U is called an “isolating neighborhood of C”.
Proposition 6.4.5
If \(C\subseteq X\) is \(\sigma \)-invariant, U is a compact neighborhood of C and \(C=I(U)\subseteq \mathrm{int}\, C\), then \(U\in \Sigma \).
Proof
Arguing indirectly, suppose that for every \(n\in \mathbb N\), we have
So, we can find \(u_n\in G^n(U)\backslash \mathrm{int}\, U\). We have
Due to the compactness of U, we may assume that
a contradiction. \(\square \)
Proposition 6.4.6
If \(K\in \Sigma \), then there exists a \(b>0\) such that \(G^b(K)\in \Sigma \) and for all \(t\in \mathbb R,\ \sigma (t, K)\in \Sigma \).
Proof
Both sets \(G^b(K)\) and \(\sigma (t, K)\) are closed (see Remark 6.4.2).
Since \(K\in \Sigma \), we can find \(b>0\) such that \(G^b(K)\subseteq \mathrm{int}\, K\). Using Lemma 6.4.3 we have
Also, we have
The proof is now complete.\(\square \)
Proposition 6.4.7
If \(K\in \Sigma \), and \(b>0\), then \(\Gamma ^b(K)\) is closed and \(\Gamma ^b(K)\subseteq \partial G^b(K)\).
Proof
Let \(\{u_n\}_{n\geqslant 1}\subseteq \Gamma ^b(K)\) and suppose that \(u_n\rightarrow u\). Then we can find \(t_n\in [0,b]\) such that \(\sigma (t_n, u_n)\in \partial K\) for all \(n\in \mathbb N\). We may assume that \(t_n\rightarrow t\in [0,b]\). We have
Next, let \(u\in \Gamma ^b(K)\). We can find \(t\in [0,b]\) and \(v_n\in X\backslash K,\ n\in \mathbb N\), such that
Let \(y_n=\sigma (-t, v_n),\ n\in \mathbb N\). Then from the semigroup property (see Definition 6.4.1), we have
Since the set \(G^b(K)\) is closed, we have \(u\notin \mathrm{int}\, G^b(K)\) and so we conclude that \(u\in \partial G^b(K)\). Therefore
The proof is now complete. \(\square \)
We take the next definition from the theory of dynamical systems.
Definition 6.4.8
Let (X, d) be a metric space and \(\sigma (t, x)\) a flow on it. For every \(u\in X\) the set
is called the “\(\omega \)-limit set of u”. The set
is called the “\(\omega ^*\)-limit set of u”. Also, given a set \(S\subseteq X\), the “\(\sigma \)-invariant hull of S” is defined to be the set
Remark 6.4.9
Note that \(\omega (u)=\omega (\sigma (t, u))\) and \(\omega ^*(u)=\omega ^*(\sigma (t, u))\) for all \(t\in \mathbb R\). So, it is clear that [S] is \(\sigma \)-invariant and is the minimal \(\sigma \)-invariant set containing S. If S is \(\sigma \)-invariant, then \(S\subseteq [S]\). Finally, these limit sets are described equivalently by
Both sets are closed and \(\sigma \)-invariant.
Next we present two important flows with useful invariant sets.
Example 6.4.10
(a) Let X be a Banach space and \(g:X\rightarrow X\) a compact map. Let \(\varphi =\mathrm{id}_X-g\) and consider the abstract Cauchy problem
Then \(\sigma (t, u)\) is a flow on X and any subset of the fixed point set of g is \(\sigma \)-invariant.
(b) Let X be a Banach space and suppose \(\varphi \in C^1(X)\) satisfies the PS-condition. Consider a pseudogradient vector field \(V(\cdot )\) of \(\varphi \). Let \(g(u)=\min \{d(u,K_{\varphi }), 1\}\) and consider the abstract Cauchy problem.
Then \(\sigma (t, u)\) is a flow on X and any subset of \(K_{\varphi }\) is \(\sigma \)-invariant.
Motivated by Example 6.4.10(b), we make the following definition.
Definition 6.4.11
A triple \((X,\varphi ,\sigma )\) is a “pseudogradient flow”, if X is a Banach space, \(\varphi \in C^1(X)\) and satisfies the PS-condition and \(\sigma (t, u)\) is the flow generated by (6.84).
Proposition 6.4.12
If \((X,\varphi ,\sigma )\) is a pseudogradient flow, then for any \(u\in X\), the sets \(\omega (u),\omega ^*(u)\) are compact and
for some critical values \(c, c^*\in \mathbb R\).
Proof
We prove the statement for \(\omega (u)\), the proof for \(\omega ^*(u)\) being similar.
We claim that \(\omega (u)\) is located in one level set of \(\varphi \), that is,
To see this, we argue by contradiction. So, suppose we could find \(\{t_n\}_{n\geqslant 1}\) and \(\{s_n\}_{n\geqslant 1}\) in \((0,+\infty )\) such that \(t_n, s_n\uparrow +\infty \) and
Without any loss of generality we may assume that \(t_n<s_n\) for all \(n\in \mathbb N\). Recalling that the pseudogradient flow is \(\varphi \)-decreasing (see the proof of Theorem 5.3.7), we have
Next we show that \(\omega (u)\subseteq K_{\varphi }\) and this, combined with the first part of the proof, implies that \(\omega (u)\subseteq K^c_{\varphi }\). Again we argue by contradiction. So, suppose we can find \(h\in \omega (u)\backslash K_{\varphi }\). Then choose regular values \(a<b\) such that \(\varphi (u),\varphi (h)\in (a, b)\). The set \(K^{[a, b]}_{\varphi }=K_{\varphi }\cap \varphi ^{-1}([a, b])\) is compact. So, we can find \(r>0\) such that
Since \(h\in \omega (u)\), from Remark 6.4.9, we know that there exists a sequence \(t_n\rightarrow +\infty \) such that \(u_n=\sigma (t_n, h)\in B_r(h)\) for all \(n\in \mathbb N\).
There exists a sequence \(s_n\rightarrow +\infty \) such that \(y_n=\sigma (s_n,h)\in \partial (K^{[a, b]}_{\varphi })_r\), where
(the r-neighborhood of \(K^{[a, b]}_{\varphi }\)). Indeed, if this is not true, then we can find \(d>0\) such that
Recall that \(\varphi \) satisfies the PS-condition. So, we can find \(\eta >0\) such that
Then
a contradiction to the choice of a.
Now choose \(\tau _n\rightarrow +\infty \) with \(s_n<\tau _n\) for all \(n\in \mathbb N\) such that
From the mean value theorem, we have
which is impossible since \(\omega (u)\) is located in one level set of \(\varphi \). \(\square \)
We introduce a notion concerning a flow \(\sigma (t, u)\), which is critical in our analysis.
Definition 6.4.13
Let (X, d) be a metric space and \(\sigma (t, u)\) a flow on X. A set \(D\subseteq X\) is said to have the “mean value property” (MVP for short) for the flow \(\sigma \) if for all \(u\in X\) and all \(t_1<t_2\)
Using this definition, we have the following auxiliary result.
Proposition 6.4.14
If \((X,\varphi ,\sigma )\) is a pseudogradient flow, S a critical set of \(\varphi \) (that is, \(S\subseteq K_{\varphi }\)) and D is a closed neighborhood of S with the MVP such that
then
-
(a)
\(I(D)=[S]\subseteq \mathrm{int}\, D;\)
-
(b)
for any \(t_1<0<t_2\), the set \(G^{t_2}_{t_1}(D)\) is a closed neighborhood of [S] with the MVP.
Proof
(a) We first show that
To this end let \(u\in [S]\). Then by Definition 6.4.8, we have
So, according to Remark 6.4.9 we can find \(t^+_n\rightarrow +\infty \) and \(t^-_n\rightarrow -\infty \) as \(n\rightarrow \infty \) such that
Because D has the MVP, it follows that
Next we show that
Let \(u\in I(D)\). Then \(\sigma (t, u)\in D\) for all \(t\in \mathbb R\). Recall that \(D\subseteq X\) is closed. So, from Remark 6.4.9 if follows that
Also, from Proposition 6.4.12, we have
From (6.87) and (6.88), we have
So, we have proved (6.86). From (6.85) and (6.86), we conclude that
Next, we show that \([S]\subseteq \mathrm{int}\, D\). So, let \(u\in [S]\). There exist \(t^-<0<t^+\) and neighborhoods \(U^{\pm }\) of \(h^{\pm }=\sigma (t^{\pm }, u)\) such that \(U^{\pm }\subseteq D\). Let
Evidently, V is a neighborhood of u such that
The MVP of D implies that \(V\subseteq D\). Therefore we conclude that
(b) From its definition it is clear that the set \(G^{t_2}_{t_1}(D)\) is closed and has the MVP. We have
We need to show that \([S]\subseteq \mathrm{int}\, G^{t_2}_{t_1}(D)\). Arguing by contradiction, suppose that we can find \(u\in [S]\cap \partial (G^{t_2}_{t_1}(D))\). So, we can find a sequence \(\{u_n\}_{n\geqslant 1}\) such that
Therefore we can find \(t_n\in [t_1,t_2]\) such that \(\sigma (t_n, u_n)\notin D\) for all \(n\in \mathbb N\). We may assume that \(t_n\rightarrow t\). Then \(\sigma (t_n,u_n)\rightarrow \sigma (t, u)\) (see Definition 6.4.1) and \(\sigma (t,u)\notin \mathrm{int}\, D\). But \(u\in [S]\) and so from Definition 6.4.8 (see also Remark 6.4.9), we have
a contradiction. \(\square \)
Proposition 6.4.15
If \((X,\varphi ,\sigma )\) is a pseudogradient flow, S is a critical set of \(\varphi \) (that is, \(S\subseteq K_{\varphi }\)) and D is a closed neighborhood of S with the MVP such that
with \(a, b\in \mathbb R\) being regular values of \(\varphi \), then for any neighborhood U of S, we can find \(c>0\) such that
Proof
We will show the contrapositive. Namely, we will show that if \(u\notin \mathrm{int}\, U\), then we can find \(c>0\) and \(t\in [-c, c]\) such that \(\sigma (t, u)\notin D\).
Since \(\varphi \) satisfies the PS-condition, we can find \(\eta >0\) such that
Choose \(c>\frac{1}{\eta _2}(b-a)\). We consider three distinct cases:
-
(1)
\(u\notin D\): Let \(t=0\). We have \(\sigma (0,u)=u\notin D\).
-
(2)
\(u\in D\backslash (\mathrm{int}\, U)^{\infty }\) (recall \((\mathrm{int}\, U)^{\infty }={\underset{\mathrm {t\in \mathbb R}}{\bigcup }}\sigma (t, U)\)): Suppose that for \(c>0\)
$$\begin{aligned}&\sigma ([-c,c],u)\subseteq D\\\Rightarrow & {} \sigma ([-c,c],u)\subseteq D\backslash (\mathrm{int}\,U)^{\infty }\\\Rightarrow & {} \varphi (\sigma (-c,u))-\varphi (\sigma (c, u))\geqslant 2\eta ^2c>b-a\ (\text {see (6.89)}), \end{aligned}$$which contradicts the hypothesis that \(D\subseteq \varphi ^{-1}([a, b])\).
-
(3)
\(u\in [(\mathrm{int}\,U)^{\infty }\backslash \mathrm{int}\,U]\cap D=[(\mathrm{int}\,U)^{\infty }\cap D]\backslash \mathrm{int}\, U\): Then
$$(3)_i\ u\in {\underset{\mathrm {t>0}}{\bigcup }}\sigma (t,\mathrm{int}\, U)\ \text {or}\ (3)_{ii}\ u\in {\underset{\mathrm {t<0}}{\bigcup }}\sigma (t,\mathrm{int}\, U).$$If \((3)_i\) holds, then we can find \(t_1\leqslant 0\leqslant t_2\) such that
$$\sigma ([t_1,t_2],u)\subseteq [(\mathrm{int}\,U)^{\infty }\cap D]\backslash \mathrm{int}\, U\ \text {and}\ \sigma (t_1-\epsilon , u)\in U,\sigma (t_2+\epsilon , u)\notin D,$$for \(\epsilon >0\) small. Then
$$\begin{aligned}&b-a\geqslant \eta ^2(t_2-t_1)\\\Rightarrow & {} t_2<c. \end{aligned}$$If \((3)_{ii}\) holds, then in a similar fashion we show that \(t_1>-c\). So, for both \((3)_i\) and \((3)_{ii}\) we have
$$\sigma ([-c,c], u)\cap D^c\ne \emptyset .$$
The proof is now complete.\(\square \)
Now we are ready for our first theorem.
Theorem 6.4.16
If \((X,\varphi ,\sigma )\) is a pseudogradient flow, S is a critical set of \(\varphi \), that is, \(S\subseteq K_{\varphi }\) and D is a closed neighborhood of S with the MVP such that
then [S] is an isolated \(\sigma \)-invariant set and any closed neighborhood U of [S] with \(U\subseteq D\) is an isolating neighborhood of [S].
Proof
From Proposition 6.4.14, we have
Let U be a closed neighborhood of [S] such that \(U\subseteq D\). Then
By definition
So, we can use Proposition 6.4.15 and find \(c>0\) such that
The proof is now complete.\(\square \)
We introduce the fundamental notion of “dynamically isolated critical set”.
Definition 6.4.17
Let \((X,\varphi ,\sigma )\) be a pseudogradient flow and S a critical set of \(\varphi \) (that is, \(S\subseteq K_{\varphi }\)). We say that S is a “dynamically isolated critical set” if there exists a closed neighborhood D of S and regular values \(a<b\) of \(\varphi \) such that
(recall \(D^{\infty }={\underset{\mathrm {t\in \mathbb R}}{\bigcup }}\sigma (t, D)\)). We say that (D, a, b) is an “isolating triplet” for S.
Remark 6.4.18
If C is an isolated critical value of \(\varphi \) (that is, there exists an \(\epsilon >0\) such that \([c-\epsilon , c+\epsilon ]\) contains no critical values other than c), then \(K^c_{\varphi }\) is a dynamically isolated critical set. Similarly, if \(u_0\) is an isolated critical point of \(\varphi \), then the singleton \(S=\{u_0\}\) is a dynamically isolated critical set. In particular, if \(u_0\) is a nondegenerate critical point of \(\varphi \in C^2(H)\) (H being a Hilbert space), then the singleton \(S=\{u_0\}\) is a dynamically isolated critical set.
Lemma 6.4.19
If \((X,\varphi ,\sigma )\) is a pseudogradient flow, S is a critical set of \(\varphi \) (that is, \(S\subseteq K_{\varphi }\)) and (D, a, b) is an isolating triplet for S, then there exists a \(c>0\) such that
moreover the set \(D^{\infty }\cap \varphi ^{-1}([a, b])\) is a closed neighborhood of both S and [S], which has the MVP.
Proof
Let \(V=\overline{D^{\infty }}\cap \varphi ^{-1}([a, b])\). We need to show that \(V=D^c\cap \varphi ^{-1}([a, b])\) for some \(c>0\). Since \(\varphi \) satisfies the PS-condition, we can find \(\eta >0\) such that
If \(\sigma ([0,t], u)\subseteq V\backslash D\), then
Choose \(c>\frac{1}{\eta ^2}(b-a)\). If the equality \(V=D^c\cap \varphi ^{-1}([a, b])\) fails, we can find
a contradiction (see Definition 6.4.17). The last part of the lemma follows from Theorem 6.4.16. \(\square \)
Combining Lemma 6.4.19 with Theorem 6.4.16, we obtain:
Theorem 6.4.20
If \((X,\varphi ,\sigma )\) is a pseudogradient flow and S is a dynamically isolated critical set of \(\varphi \), then [S] is an isolated \(\sigma \)-invariant set and if (D, a, b) is an isolating triplet for S, then any closed neighborhood U of [S] such that \(U\subseteq D\) is an isolating neighborhood of [S].
Now we can extend the notion of critical groups from an isolated critical point (see Definition 6.2.1) to a dynamically isolated critical set.
Definition 6.4.21
Let \((X,\varphi ,\sigma )\) be a pseudogradient flow, S a dynamically isolated critical set and (D, a, b) an isolating triplet for S. The “critical groups” of S are defined by
For this definition to make sense, we need to show that it is independent of the choice of the isolating triplet (D, a, b) and the choice of the pseudogradient vector field for \(\varphi \). This is done in the next proposition.
Proposition 6.4.22
The definition of critical groups for a dynamically isolated critical set S (see Definition 6.4.21) is independent of the particular choice of the isolating triplet and of the pseudogradient vector field.
Proof
First we assume that in the isolating triple (D, a, b), the neighborhood D is fixed and the regular values a, b vary. Then the invariance of \(C_k(\varphi , S),\ k\in \mathbb N_0\), is a consequence of Corollary 5.3.13.
Next suppose that (D, a, b) and \((D_*,a, b)\) are two isolating triplets for S such that
We need to show that
Claim 1: There exists a \(\delta >0\) such that \(d(\partial D^{\infty }_{*,+},[S])\geqslant \delta \).
If Claim 1 is not true, then we can find a sequence \(\{u_n\}_{n\geqslant 1}\) such that
So, we can find \(v_n\in \partial D_*\) and \(t_n\in [0,c]\), \(n\in \mathbb N\), such that
By passing to a suitable subsequence if necessary, we may assume that
The set \(\partial D_*\) is closed, hence \(v\in \partial D_*\). But recall that \(u\in [S]\). Hence \(v\in [S]\), a contradiction [see (6.91)]. This proves Claim 1.
Choose \(a_*<a\) so that \((D_*,b, a_*)\) is still an isolating triple for S (recall a is a regular value). We set
For each \(u\in E\), there exists a unique \(h\in A\) and a unique \(t\in \mathbb R\) such that
Let \(p:E\rightarrow A\) and \(q:E\rightarrow \mathbb R\) be the maps defined by
We set
Claim 2: \(d(C, p(\partial D^{\infty }_{+}))>0\).
If Claim 2 is not true, then we can find a sequence \(\{h_n\}_{n\geqslant 1}\) such that
So, we can find \(v_n\in \partial D\) and \(t_n\in [0,c]\) such that \(v_n=\sigma (-t_n, h_n)\) for all \(n\in \mathbb N\). We may assume that \(t_n\rightarrow t\) and so we have
But \(h\in C\) and so we have a contradiction. This proves Claim 2.
Let \(E_*=\{u\in D^{\infty }_{*,+}:\omega (u)\cap S^c\ne \emptyset \}\). From Claim 2 we have
Consider the parametric family of continuous functions \(\gamma _{\tau }:\mathbb R_+\rightarrow \mathbb R\), \(\tau \in [0,1]\), defined by
Then we introduce the deformation \(\xi :[0,1]\times (\varphi ^b\cap D^{\infty }_{+})\rightarrow \varphi ^b\cap D^{\infty }_{+}\) defined by
where \(h=p(u)\) and \(s=q(u)\). Then \(\xi (0,u)=u\) and
Let \(L_1=\xi (1,\varphi ^b\cap D^{*}_{+})\) and \(L_2=L_1\backslash (A^{\infty }_+\backslash A_1)\). Using the properties of relative singular homology groups (see Sect. 6.1), we have
In a similar fashion, we also show the invariance with respect to the pseudogradient vector field. \(\square \)
To describe the topological properties of a dynamically isolated critical set, we will need the following notion.
Definition 6.4.23
Let \((X,\varphi ,\sigma )\) be a pseudogradient flow and S a critical set of \(\varphi \) (that is, \(S\subseteq K_{\varphi }\)). A pair of sets \((W, W_-)\) is said to be a “Gromoll–Meyer pair” (a GM-pair for short) associated with the pseudogradient flow if the following conditions hold:
-
(a)
W is a closed neighborhood of S with the MVP such that
$$W\cap K_{\varphi }=S\ \text {and}\ W\cap \varphi ^a=\emptyset \ \text {for some}\ a\in \mathbb R.$$ -
(b)
\(W_-\) is an exit set of W, that is, for every \(u\in W\) and \(t_1>0\) such that \(\sigma (t_1,u)\notin W\), we can find \(\hat{t}\in \left[ 0,t_1\right) \) for which we have
$$\sigma ([0,\hat{t}],u)\subseteq W\ \text {and}\ \sigma (\hat{t}, u)\in W_-.$$ -
(c)
\(W_-\) is closed and is the union of a finite number of submanifolds which are transversal to the flow \(\sigma \).
Example 6.4.24
In this example, we construct a GM-pair for an isolated critical point. So, let \(X=H\) be a Hilbert space and suppose \(\varphi \in C^1(H)\) satisfies the PS-condition. For simplicity we assume that the critical point \(u_0=0\) and that \(\varphi (0)=0\). Let \(\epsilon ,\delta >0\) such that
The PS-condition implies that
Let \(\lambda \in \left( 0,\frac{2\delta }{\eta }\right) \) and consider the functional
We choose \(\gamma ,\mu >0\) in such a way that if \(W=\psi ^{\mu }\cap \varphi ^{-1}([-\gamma ,\gamma ])\) and \(W_-=W\cap \varphi ^{-1}(-\gamma )\), then the following conditions hold
We claim that \((W, W_-)\) is a GM-pair.
First we show that W has the MVP. There is no loss of generality if we assume that \(\sigma (0,u),\sigma (t, u)\in W\ u\in W\). Let \(t_0=\sup \{s\in [0,t]:\sigma (s', u)\in W\ \text {for all}\ u\leqslant s'\leqslant s\}\). If \(t_0<t\), then \(\sigma (t_0,u)\notin B_{\delta /2}(0)\). But we have
From (6.97) and (6.98) we have a contradiction to the maximality of \(t_0\).
Let \(\tilde{W}_-=\{u\in W:\sigma (t, u)\notin W\ \text {for all}\ t>0\}\). Evidently,
By definition, \(\tilde{W}_-\subseteq \partial W=W_-\cup (\varphi ^{-1}(\gamma )\cap \mathrm{int}\psi ^{\mu })\cup (\psi ^{-1}(\mu )\cap (W\backslash W_-))\). If \(u\in \varphi ^{-1}(\gamma )\cap \mathrm{int}\,\psi ^{\mu }\), then \(u\notin W_-\). If \(u\in \psi ^{-1}(\mu )\cap (W\backslash W_-)\), then from (6.95) and (6.96) we have
So, we can find \(\tau >0\) such that
From Definition 6.4.23 it follows that \((W, W_-)\) is a GM-pair.
We can extend this example from an isolated critical point to a dynamically isolated critical set.
Theorem 6.4.25
If \((X,\varphi ,\sigma )\) is a pseudogradient flow, S is a dynamically isolated critical set and (D, a, b) is an isolating triplet for S, then for any neighborhood U of [S] such that
there exists a GM-pair \((W, W_-)\) for S such that \(W\subseteq U\).
Proof
Let \(a<a_0<\min [\varphi (u):u\in S]\). From Lemma 6.4.19 we know that \(D^{\infty }\cap \varphi ^{-1}([a_0,b])\) is a closed neighborhood of [S] which has the MVP. From Proposition 6.4.15, we know that there exists a \(c>0\) such that
Proposition 6.4.14 implies that
Moreover, we have
We look for an exit set E of W. Let \(L_{a_0}=D^{\infty }\cap \varphi ^{-1}(a_0)\). This is a submanifold of \(\varphi ^{-1}(a_0)\).
Since W is a neighborhood of S, we have
So, for all \(u\in E\), we can find \(t>0\) such that
The proof is now complete.\(\square \)
Theorem 6.4.26
If \((X,\varphi ,\sigma )\) is a subgradient flow and S is a dynamically isolated critical set of \(\varphi \), then for any GM-pair for S we have
Proof
Let (D, a, b) be an isolating triplet for S. Using Theorem 6.4.25, we replace D by W. First we show that
(recall that \(W^{\infty }_+={\underset{\mathrm {t\geqslant 0}}{\bigcup }}\sigma (t, W)\ \text {and}\ (W_-)^{\infty }_+={\underset{\mathrm {t\geqslant 0}}{\bigcup }}\sigma (t, W_-)\)).
We introduce two deformation retractions
defined as follows. Let \(\vartheta _1:(W_-)^{\infty }_+\rightarrow \mathbb R\) be the first hitting time from \((W_-)^{\infty }_+\) at the level set \(\varphi ^{-1}(a)\). So, we have
The transversality of \(\sigma \) to \(\varphi ^{-1}(a)\) implies the continuity of \(\vartheta _1\).
Similarly, let \(\vartheta _2:W^{\infty }_+\rightarrow \mathbb R\) be the first hitting time from \(W^{\infty }_+\) at the level set \(\varphi ^{-1}(b)\). So, as before we have
For the same reason \(\vartheta _2\) is continuous.
We set
Since \(W\cap \varphi ^a=\emptyset \) (see Definition 6.4.23), we have
Then using the deformation retractions \(d_1\) and \(d_2\), we have that
are homotopy equivalent. Hence by Proposition 6.1.14 we have (6.100).
Next we show that
Let \(\delta >0\) and set \(W_{\delta }={\underset{\mathrm {t>\delta }}{\bigcup }}\sigma (t, W_-)\). We consider \(\vartheta :W^{\infty }_+\rightarrow \mathbb R\), the first hitting time at the set \(W_-\). We have
Recall that the flow \(\sigma \) is transversal to \(W_-\) [see Definition 6.4.23(c)]. So \(\vartheta (\cdot )\) is continuous. Also, we have
and so \(W_{\delta }\) is relatively open in \(W^{\infty }\). We have
The excision property of singular homology implies that
Let \(d_3:[0,1]\times (W^{\infty }\backslash W_{\delta })\rightarrow W\) and \(d_4:[0,1]\times ((W_-)^{\infty }_+\backslash W_{\delta })\rightarrow W\) be the deformations defined by reversing the flow, that is,
These are strong deformation retractions. So, we have
From (6.100), (6.101), (6.102), (6.103) and invoking Proposition 6.4.22, we conclude that
The proof is now complete.\(\square \)
We want to study the stability of the critical groups for dynamically isolated critical sets under perturbations of the flow (therefore under changes in the pseudogradient vector field).
So, let \((S,\varphi ,\sigma _{\lambda })\ \lambda \in [0,1]\), be a family of pseudogradient flows. We impose the following “uniform continuity condition”.
(UC) : “For every \(\epsilon >0\) and \(b>0\), there exists a \(\delta (\epsilon , b)>0\) such that
imply
The following proposition is an immediate consequence of this uniformity condition.
Proposition 6.4.27
If \((X,\varphi ,\sigma _{\lambda }),\ \lambda \in [0,1]\), is a family of pseudogradient flows which satisfies the (UC), S is a dynamically isolated critical set of the flow \(\sigma _{\lambda _0}\) and (D, a, b) is an isolating triplet for S, then there exists a \(\delta >0\) such that
Remark 6.4.28
In fact the isolating neighborhood of S (see Definition 6.4.4(b)) is also stable under small changes of the parameter \(\lambda \in [0,1]\).
Next we examine the effect on GM-pairs when we perturb the functional \(\varphi \).
Theorem 6.4.29
If \((X,\varphi ,\sigma )\) is a pseudogradient flow, \(S_{\varphi }\) is critical set of \(\varphi \) and \((W, W_-)\) is a GM-pair for \(S_{\varphi }\), then there exists an \(\epsilon =\epsilon (\varphi , W)>0\) such that for all \(\psi \in C^1(X)\) with
\((W, W_-)\) is still a GM-pair for any critical set \(S_{\psi }\) of \(\psi \) such that
Proof
Let U be a neighborhood of \(S_{\varphi }\) such that \(U\subseteq \bar{U}\subseteq \mathrm{int}\, W\). Let V be the pseudogradient vector field associated with the flow \((X,\varphi ,\sigma )\). Since \(\varphi \) satisfies the PS-condition (see Definition 6.4.11), we can find an open neighborhood \(U_0\) of \(S_{\varphi }\) such that
Let \(\epsilon \in (0,\frac{\eta }{6})\) and let \(\psi \in C^1(X)\) be such that
Evidently, \(S_{\psi }\subseteq U_0\).
Consider a pseudogradient vector field \(\hat{V}\) for \(\psi \) such that
Let \(\vartheta :X\rightarrow \mathbb R\) be a locally Lipschitz function such that
Let \(V_0(u)=\frac{5}{4}\left[ (1-\vartheta (u))V(u)+\vartheta (u)\hat{V}(u)\right] \) for all \(u\in X\). For \(u\in X\backslash U_0\) we have
If \(u\in U_0\), then \(V_0(u)=\hat{V}(u)\). Therefore \(V_0\) is also a pseudogradient vector field for \(\psi \).
Note that \(V_0(u)=\frac{5}{4}V(u)\) for all \(u\in X\backslash U\). Therefore the flow \(\sigma _0\) corresponding to \(V_0\) remains the same as the flow \(\sigma \). In particular, they are the same on \(W_-\). Also, we can easily check that W satisfies the MVP for the flow \(\sigma _0\). Therefore \((W, W_-)\) remains a GM-pair for \(S_{\psi }\). \(\square \)
Definition 6.4.30
Let \((X,\varphi ,\sigma )\) be a pseudogradient flow and S a critical set of \(\varphi \) (that is, \(S\subseteq K_{\varphi }\)). A subset \(A\subseteq [S]\) is called an “attractor” in [S] if there is a neighborhood U of A such that \(\omega (U\cap [S])=A\). The dual “repeller” of A in [S] is defined by \(A^*=\{u\in [S]:\omega (u)\cap A=\emptyset \}\). The pair \((A, A^*)\) is said to be an “attractor-repeller pair”. An ordered collection \(\{M_k\}^n_{k=1}\) of \(\sigma \)-invariant subsets \(M_k\subseteq [S]\) is said to be a “Morse decomposition” of [S] if there is an increasing family of attractors
such that \(M_k=A_k\cap A^*_{k-1}\) for all \(k\in \{1,\ldots , n\}\).
Remark 6.4.31
An attractor-repeller pair \((A, A^*)\) of [S] is a Morse decomposition with \(A_0=\emptyset , A_1=A, A_2=[S]\). More generally, suppose that \(a, b\in \mathbb R\) are regular values of \(\varphi \) and assume that \(\varphi ^{-1}([a, b])\cap K_{\varphi }=\{u_k\}^n_{k=1}\) with \(\varphi (u_k)\leqslant \varphi (u_{k+1})\) for all \(k\in \{1,\ldots , n-1\}\). Then \(\{\{u_k\}\}^n_{k=1}\) is a Morse decomposition of \([S]=I(\varphi ^{-1}([a, b]))\).
Then we can have an extension of the Morse relation from Theorem 6.2.20. For a proof of this result, we refer to Chang [118] (Sect. 5.5).
Theorem 6.4.32
If \((X,\varphi ,\sigma )\) is a pseudogradient flow, S is a critical set of \(\varphi \) (that is, \(S\subseteq K_{\varphi }\)), \(\{M_i\}^n_{i=1}\) is a Morse decomposition of [S] and \((W, W_-)\) is a GM-pair for [S], then \({\underset{\mathrm {k\in \mathbb N_0}}{\sum }}\left( \overset{n}{\underset{\mathrm {i=1}}{\sum }}\mathrm{rank}\, H_k(W_i, W_{i-1})t^k\right) ={\underset{\mathrm {k\in \mathbb N_0}}{\sum }}\mathrm{rank}\,H_k(W, W_-)t^k+(1+t)Q(t)\), where \((W_i, W_{i-1})\) is the GM-pair for \(M_i,\ i\in \{1,\ldots , n\}\) and Q(t) is a formal series with nonnegative integer coefficients.
Remark 6.4.33
Suppose that \([S]=\{u_i\}^n_{i=1}\). From Remark 6.4.31 we know that \(\{u_i\}^n_{i=1}\) is a Morse decomposition of [S]. We set
We assume that they are all finite and that the series that we are about to formally introduce converge. We have
These are the extended Morse relations.
6.5 Local Extrema and Critical Points of Mountain Pass Type
The main idea of Morse theory is that different critical points of a functional \(\varphi \in C^1(X)\) can be distinguished by the topological structure of their neighborhoods in the sublevel sets of \(\varphi \). In fact such topological information can also be extracted from the minimax characterization of the corresponding critical values.
We start with some easy observations concerning local extrema.
In Proposition 6.2.3, we saw that if \(u_0\) is a local minimum of \(\varphi \), then
In the next proposition we complete this result.
Proposition 6.5.1
If X is a reflexive Banach space, \(\varphi \in C^1(X)\), \(\varphi \) satisfies the C-condition, \(u_0\in K_{\varphi }\) is isolated and \(c_0=\varphi (u_0)\) is isolated in \(\varphi (K_{\varphi })\), then the following statements are equivalent:
-
(a)
\(u_0\) is local minimizer of \(\varphi \);
-
(b)
\(C_k(\varphi , u_0)=\delta _{k, 0}\mathbb Z\) for all \(k\in \mathbb N_0\);
-
(c)
\(C_0(\varphi , u_0)\ne 0\).
Proof
\((a)\Rightarrow (b):\) This implication is Proposition 6.2.3.
\((b)\Rightarrow (c):\) Obvious.
\((c)\Rightarrow (a):\) Arguing by contradiction suppose that \(u_0\) is not a local minimizer of \(\varphi \). By Lemma 6.2.35, we can find \(a, b\in \mathbb R\) such that
Then Definition 6.2.1, Theorem 5.3.12 and Corollary 6.1.24 imply that
Let \(h:[0,1]\times \varphi ^b\rightarrow \varphi ^b\) be the deformation into \(\varphi ^{c_0}\) provided by Theorem 5.3.12 (the second deformation theorem). Then for any \(u\in \varphi ^b,\ h(\cdot , u)\) is a path in \(\varphi ^b\) which connects u to \(b(1,u)\in \varphi ^{c_0}\).
Next note that we can find a small \(r>0\) such that \(\varphi (y)<b\) for all \(y\in B_r(u_0)=\{v\in X:||v-u_0||<r\}\). Since by hypothesis \(u_0\) is not a local minimizer of \(\varphi \), we can find \(\hat{u}\in B_r(u_0)\) such that \(\varphi (\hat{u})<c_0\). Then \(\gamma (t)=(1-t)u_0+t\hat{u}\), \(t\in [0,1]\) is a path connecting \(u_0\) and \(\hat{u}\) and staying in \(\varphi ^b\).
So, we have seen that every element \(u\in \varphi ^b\) can be connected to an element of \(\varphi ^{c_0}\backslash \{u_0\}\) by a path staying in \(\varphi ^b\). Then according to Remark 6.1.50 we have \(C_0(\varphi , u_0)=H_0\left( \varphi ^b,\varphi ^{c_0}\backslash \right. \) \(\left. \{u_0\}\right) =0\). So, \(u_0\) is a local minimizer of \(\varphi \). \(\square \)
Combining Proposition 6.5.1 and Lemma 6.2.38, we obtain:
Proposition 6.5.2
If \(\varphi \in C^2(\mathbb R^N)\) and \(u_0\in K_{\varphi }\) is isolated, then the following statements are equivalent:
-
(a)
\(u_0\) is a local maximizer of \(\varphi \);
-
(b)
\(C_k(\varphi , u_0)=\delta _{k, 0}\mathbb Z\) for all \(k\in \mathbb N_0\);
-
(c)
\(C_{N}(\varphi , u_0)\ne 0\).
In fact a similar result holds for \(C^2\)-functions on \(\mathbb R^N\) and local maximizers (see Mawhin and Willem [293, p. 193]).
Proposition 6.5.3
If \(\varphi \in C^2(\mathbb R^N)\) and \(u_0\in K_{\varphi }\) is isolated, then the following statements are equivalent:
-
(a)
\(u_0\) is local minimizer of \(\varphi \);
-
(b)
\(C_k(\varphi , u_0)=\delta _{k, N}\mathbb Z\) for all \(k\in \mathbb N_0\);
-
(c)
\(C_n(\varphi , u_0)\ne 0\).
Remark 6.5.4
So, if \(\varphi \in C^2(\mathbb R^N)\) and \(u\in K_{\varphi }\) is isolated and it is neither a local minimizer or a local maximizer, then \(C_0(\varphi , u_0)=C_N(\varphi , u_0)=0\).
Also, as a consequence of Lemma 6.2.38, we have:
Proposition 6.5.5
If \(\varphi \in C^2(\mathbb R^N)\) and \(u_0\in K_{\varphi }\) is isolated, then \(\mathrm{rank}\,C_k(\varphi , u_0)\) is finite for all \(k\in \mathbb N_0\) and \(C_k(\varphi , u_0)=0\) for all \(k\notin \{0,1,\ldots , N\}\).
Next we recall a notion from Sect. 5.7 (see Definition 5.7.2):
Definition 6.5.6
Let X be a Banach space, \(\varphi \in C^1(X)\) and \(u\in K_{\varphi }\). We say that u is of “mountain pass type” if for any open neighborhood U of u, the set \(\{v\in U:\varphi (v)<\varphi (u)\}\) is nonempty and not path connected.
Remark 6.5.7
In Theorem 5.7.7, we established that if \(\varphi \in C^1(X)\) satisfies the C-condition and the mountain pass geometry and \(K_{\varphi }\) is discrete, then we can find \(u\in K^c_{\varphi }\) which is of mountain pass type (recall \(c=\inf \limits _{\gamma \in \Gamma }\max \limits _{0\leqslant t\leqslant 1}\varphi (\gamma (t))\), see Theorem 5.4.6).
In the next theorem we establish a useful property of the critical groups of a \(u\in K_{\varphi }\) which is of mountain pass type.
Theorem 6.5.8
If X is a reflexive Banach space, \(\varphi \in C^1(X),\ u_0\in K_{\varphi }\) is isolated, \(c_0=\varphi (u_0)\) is isolated in \(\varphi (K_{\varphi })\) and \(u_0\) of mountain pass type, then \(C_1(\varphi , u_0)\ne 0\).
Proof
Let \(\psi \in C^1(X)\) as postulated by Lemma 6.2.35. We know that \(\varphi \leqslant \psi \) and \(\varphi |_{U}=\psi |_{U}\) with U some open neighborhood of \(u_0\).
Claim 1. \(u_0\) is a critical point of mountain pass type for the functional \(\psi \).
From Lemma 6.2.35 we know that \(K_{\varphi }=K_{\psi }\) and so \(u_0\in K_{\psi }\). Let V be an open neighborhood of \(u_0\) and define
Then we have
By hypothesis, \(u_0\in K_{\varphi }\) is of mountain pass type. So, from (6.104) and Definition 6.5.6 we have that
This proves Claim 1.
Using Lemma 6.2.35(d) and Claim 1, without any loss of generality we may assume that there are \(a, b\in \mathbb R\) such that
Let C be the connected component of \(U=\varphi ^{-1}((a, b))\) which contains \(u_0\in K_{\varphi }\). Then C is open, path-connected, and contains \(u_0\) and \(K_{\varphi }\cap C=\{u_0\}\). Therefore from Definition 6.2.1, we have
From the second deformation theorem (see Theorem 5.3.12), we can find a deformation \(\hat{h}:[0,1]\times \varphi ^b\rightarrow \varphi ^b\) of \(\varphi ^b\) into \(\varphi ^{c_0}\) with the properties provided by Theorem 5.3.12. We have
So, it follows that \(\hat{h}:[0,1]\times C\rightarrow C\) is a deformation into \(C\cap \varphi ^{c_0}\) and this means that \(C\cap \varphi ^{c_0}\) is a strong deformation retract of C. Then from (6.105) and Corollary 6.1.24(b), we have
Using Axiom 4 in Definition 6.1.12 we have the exact sequence
Let \(C_0=\{u\in C:\varphi (u)<c_0\}\), \(d\in (a, c_0)\) and \(\hat{h}:[0,1]\times (\varphi ^{c_0}\backslash \{u_0\})\rightarrow \varphi ^{c_0}\backslash \{u_0\}\) be the deformation into \(\varphi ^d\) provided by the second deformation theorem (see Theorem 5.3.12).
Claim 2. \(\tilde{h}([0,1]\times (C\cap \varphi ^{c_0}\backslash \{u_0\}))\subseteq C\cap \varphi ^{c_0}\backslash \{u_0\}\) and \(\tilde{h}([0,1]\times C_0)\subseteq C_0\).
Take \(u\in C\cap \varphi ^{c_0}\backslash \{u_0\}\). Then \(\tilde{h}([0,1]\times \{u\})\subseteq V\), it is connected and intersects C (since it contains u). So, from the definition of C, we have
If \(u\in C_0\), then \(\varphi (\tilde{h}(t, u))\leqslant \varphi (u)<c_0\) for all \(t\in [0,1]\) (see Theorem 5.3.12). This proves Claim 2.
Then according to Claim 2, using \(\tilde{h}(t, u)\) we have deformations of
It follows that
Then from (6.106) and (6.107), we have
Since \(u_0\) is of mountain pass type, the set \(C_0\) is nonempty and not path connected. Therefore
Hence the homomorphism \(\gamma \) in (6.108) cannot be injective and so by the exactness of (6.108) it follows that \(C_1(\varphi , u_0)\ne 0\). \(\square \)
In the context of Hilbert spaces and of \(C^2\)-functionals, we can improve Remark 6.5.7.
Proposition 6.5.9
If H is a Hilbert space, \(\varphi \in C^2(H)\), \(u_0\in K_{\varphi }\) is isolated with finite Morse index \(m_0=m(u_0)\) and finite nullity \(\nu _0=\nu (u_0)=\mathrm{dim\, ker}\,\varphi ''(u_0)\), when \(m_0=0\) we have \(\nu _0\in \{0,1\}\) and \(C_1(\varphi , u_0)\ne 0\), then \(C_k(\varphi , u_0)=\delta _{k, 1}\mathbb Z\) for all \(k\in \mathbb N_0\).
Proof
Let \(\hat{\varphi }\in C^2(W)\) be as postulated by Proposition 6.2.9, with \(W\subseteq \mathrm{ker}\,\varphi ''(u_0)\) a neighborhood of the origin. Then from Theorem 6.2.13 (the shifting theorem), we have
Since by hypothesis \(C_1(\varphi , u_0)\ne 0\), it follows that \(m_0\in \{0,1\}\).
Case 1. \(m_0=1\).
Then from (6.109) we have \(C_0(\hat{\varphi }, 0)\ne 0\) and so Proposition 6.5.2 implies that
Case 2. \(m_0=0\).
Then from (6.109) and the hypothesis we have
By hypothesis \(\mathrm{dim\, ker}\,\varphi ''(u_0)\leqslant 1\). Then from Proposition 6.2.5 we have
The proof is now complete. \(\square \)
Remark 6.5.10
The hypotheses of the above proposition imply that \(\varphi ''(u_0)\) is a Fredholm operator (see Definition 6.2.7).
Finally, combining Theorem 6.5.8 and Proposition 6.5.9, we can state the following theorem.
Theorem 6.5.11
If H is a Hilbert space, \(\varphi \in C^2(H)\), \(\varphi \) satisfies the C-condition, \(u_0\in K_{\varphi }\) is isolated and so is \(c_0=\varphi (u_0)\) in \(\varphi (K_{\varphi })\), the Morse index \(m_0=m(u_0)\) and the nullity \(\nu _0=\nu (u_0)=\mathrm{dim\, ker}\,\varphi ''(u_0)\) are finite and \(m_0=0\) implies \(\nu _0\in \{0,1\}\) and \(u_0\) is of mountain pass type, then \(C_k(\varphi , u_0)=\delta _{k, 1}\mathbb Z\) for all \(k\in \mathbb N_0\).
6.6 Computation of Critical Groups
In this section we use Morse theory to compute the critical groups at certain particular critical points.
The notion of linking sets introduced in Definition 5.4.1 played a central role in the minimax theory of the critical values of a functional \(\varphi \in C^1(X)\). Next, we introduce an analogous notion, which will help us to produce pairs of sublevel sets with nontrivial homology groups. From Proposition 6.2.15 we know that such pairs lead to a critical level between them.
Definition 6.6.1
Let X be a Banach space and \(E_0,E, D\) nonempty subsets of X such that
We say that the pair \(\{E_0,E\}\) “homologically links” D in dimension m if the homomorphism \(i_*:H_m(E, E_0)\rightarrow H_m(X, X\backslash D)\) induced by the corresponding inclusion of \((E, E_0)\) into \((X, X\backslash D)\) is nontrivial.
Remark 6.6.2
(a) In the literature, the notion of linking introduced in Definition 5.4.1 is often called “homotopical linking” in order to distinguish it from the above concept of linking, called “homological linking”.
(b) For every \(m\in \mathbb N_0\) and \(*\in E_0\), we have the following commutative diagram of homomorphisms
with \(j_*\) being the homomorphism induced by the corresponding inclusion map. Suppose that E is contractible. Using the long exact sequence of Proposition 6.1.14 we see that the boundary homomorphisms \(\partial _1,\partial _2\) are isomorphisms. So, it follows that
In Example 5.4.3, we introduced some triplets of sets \(\{E_0,E, D\}\) which arise in the main minimax theorems and which are homotopically linking in the sense of Definition 5.4.1. In the sequel we show that these triplets are also homologically linking.
Proposition 6.6.3
If X is a Banach space, \(u_0\in X, U\) is a bounded open neighborhood of \(u_0,\ u_1\notin \bar{U}\), \(E_0=\{u_0,u_1\}\), \(E=\{tu_0+(1-t)u_1:t\in [0,1]\}\) and \(D=\partial U\), then the pair \(\{E_0,E\}\) homologically links D in dimension 1.
Proof
Let \(j:(E_0,\{u_1\})\rightarrow (X,\backslash D,\{u_1\})\) be the inclusion map and consider the map \(r:(X\backslash D,\{u_1\})\rightarrow (E_0,\{u_1\})\) defined by
Then \(r\circ j=\mathrm{id}_{(E_0,\{u_1\})}\) (here by \(\mathrm{id}_{(E_0,\{u_1\})}\) we denote the identity map seen as a map of pairs (see Definition 6.1.1(b))). Hence \(j_*:H_0(E_0,\{u_1\})\rightarrow H_0(X\backslash D,\{u_1\})\) is injective. From Example 6.1.34(b), we have
The proof is now complete.\(\square \)
Recall that for \(\rho >0\), we have
Proposition 6.6.4
If X is a Banach space, \(X=Y\oplus V\) with \(dimY<\infty \), \(E_0=\partial B_{\rho }(0)\cap Y\), \(E=\bar{B}_{\rho }(0)\cap Y\) and \(D=V\), then the pair \((E_0,E)\) homologically links D in dimension \(d=\mathrm{dim}\, Y\).
Proof
From the proof of Proposition 6.2.30, we know that
So, we have
Let \(*\in E_0\) and consider the triple \(\{*\}\subseteq E_0\subseteq X\backslash D\). Using the long exact sequence from Proposition 6.1.29, we have that \(j_*:H_{d-1}(E_0,*)=\mathbb Z\rightarrow H_{d-1}(X\backslash D,*)\) is an isomorphism, thus nontrivial. Again from (6.110) we conclude that the pair \(\{E_0,E\}\) homologically links D in dimension \(d=\mathrm{dim}\, Y\). \(\square \)
In a similar fashion, we also establish the following propositions.
Proposition 6.6.5
If X is a Banach space, \(X=Y\oplus V\) with \(\mathrm{dim},\, Y<+\infty \), \(v_0\in V\) with \(||v_0||=1\) \(0<\rho<r_1,\ 0<r_2\),
then the pair \(\{E_0,E\}\) homologically links D in dimension \(d=\mathrm{dim}\, Y+1\).
Homological linking is invariant under homeomorphisms.
Proposition 6.6.6
If X is a Banach space, the pair \(\{E_0,E\}\) homologically links D in dimension m and \(h:X\rightarrow X\) is a homeomorphism, then the pair \((h(E), h(E_0))\) homologically links h(D) in dimension m.
Proof
Just consider the following commutative diagram of homomorphisms
The proof is now complete.\(\square \)
Homological linking implies homotopical linking.
Proposition 6.6.7
If X is a Banach and the pair \((E_0,E)\) homotopically links D in dimension d, then the pair \((E_0,E)\) homologically links D (that is, in the sense of Definition 5.4.1).
Proof
Since by Definition 6.6.1, \(i_*\) is nontrivial and the homology class \([\mathrm{id}_{(E, E_0)}]\) generates \(H_m(E, E_0)\), we have that \(i_*([\mathrm{id}_{(E, E_0)}])\ne 0\) in \(H_m(X, X\backslash D)\). So, there is no relative singular homology (\(m+1\))-chain of \((X, X\backslash D)\) with boundary \(\mathrm{id}_{(E, E_0)}\). Therefore there is no map \(\gamma \in C(X, X\backslash D)\) such that \(\gamma |_{E_0}=\mathrm{id}|_{E_0}\). \(\square \)
Next we present a useful consequence of the notion of homological linking.
Proposition 6.6.8
If X is a Banach space, the pair \(\{E_0,E\}\) homologically links D in dimension m, \(\varphi \in C^1(X)\) and \(a<b\leqslant +\infty \) are such that
then
-
(a)
\(H_m(\varphi ^b,\varphi ^a)\ne \emptyset \);
-
(b)
if in addition \(\varphi \) satisfies the C-condition, \(a, b\notin \varphi (K_{\varphi })\) and \(K_{\varphi }\cap \varphi ^{-1}((a, b))\) is finite then there exists a \(u\in K_{\varphi }\cap \varphi ^{-1}((a, b))\) such that \(C_m(\varphi , u)\ne 0\).
Proof
(a) Consider the following inclusion maps of pairs of spaces
We have \(i_*=e_*\circ j_*:H_m(E, E_0)\rightarrow H_m(X, X\backslash D)\) and by hypothesis it is nontrivial. Therefore \(j_*\ne 0,e_*\ne 0\). From (6.111) it follows that \(H_m(\varphi ^b,\varphi ^a)\ne 0\).
(b) Follows from Theorem 6.2.20(b). \(\square \)
Corollary 6.6.9
If X is a Banach space, \(\varphi \in C^1(X),\ \varphi \) satisfies the C-condition, \(K_{\varphi }\) is finite, \(u_0,u_1\in X\), \(0<\rho <||u_1-u_0||\) and
then there exists a \(u\in K_{\varphi }\) with \(d\leqslant \varphi (u)\) and \(C_1(\varphi , u)\ne 0\).
Proof
Choose \(a\in (c, d)\) so that \(\left[ a, b\right) \) contains no critical values of \(\varphi \). Let
From Proposition 6.6.3 we know that the pair \(\{E, E_0\}\) homologically links D in dimension 1. Using Proposition 6.6.8(b) (with \(b=+\infty \)), we can find
Since \(\left[ a, b\right) \) contains no critical values of \(\varphi \), we must have \(\varphi (u)\geqslant d\). \(\square \)
Remark 6.6.10
This corollary is essentially Theorem 6.5.8.
Proposition 6.6.11
If X is a Banach space, \(\varphi \in C^1(X)\), the pair \(\{E, E_0\}\) homologically links D in dimension m and
then \(H_m(X,\varphi ^a)\ne 0\).
Proof
We consider the following commutative diagram of homomorphisms
By hypothesis, \(i_*\ne 0\) (see Definition 6.6.1). It follows that
Hence we have that \(i_*\) is an isomorphism and so
The proof is now complete. \(\square \)
Corollary 6.6.12
If X is a Banach space, \(\varphi \in C^1(X)\), \(\varphi \) satisfies the C-condition, the pair \(\{E, E_0\}\) homologically links D in dimension m and
then \(C_m(\varphi ,\infty )\ne 0\).
In Definition 5.4.14 we introduced the notion of local linking, which is important in many variational problems. In Remark 5.4.15 we observed that this condition implies that \(u=0\) is a critical point of the functional. So, we would like to compute its critical groups. We will do this as a consequence of our analysis of a more general notion called “homological local linking”.
Definition 6.6.13
Let X be a Banach space, \(\varphi \in C^1(X)\), \(\varphi (0)=0\) and \(0\in K_{\varphi }\) be isolated. Let \(m, n\in \mathbb N\). We say that \(\varphi \) has a “local (m, n)-linking” near the origin if there is a neighborhood U of the origin and nonempty sets \(E_0\subseteq E\subseteq U\), \(D\subseteq X\) such that \(E_0\cap D=\emptyset \) and
-
(a)
\(\varphi ^0\cap U\cap K_{\varphi }=\{0\}\);
-
(b)
\(\mathrm{rank}\, i_*-\mathrm{rank\, im}\, j_*\geqslant n\), where \(i_*:H_{m-1}(E_0)\rightarrow H_{m-1}(X\backslash D)\) and\(j_*:H_{m-1}(E_0)\rightarrow H_{m-1}(E)\) are the homomorphisms induced by the inclusions \(i:E_0\rightarrow X\backslash D\) and \(j:E_0\rightarrow E\);
-
(c)
\(\varphi |_E\leqslant 0<\varphi |_{U\cap D\backslash \{0\}}\) .
We want to know how this notion is related to the concept of local linking (see Definition 5.4.14). To do this, we will need the following deformation lemma.
Lemma 6.6.14
If X is a Banach space, \(X=Y\oplus V\) with \(d=\mathrm{dim}\, Y<+\infty \), \(\varphi \in C^1(X)\) has a local linking at 0 (with respect to the pair (Y, V)), that is, there exists an \(r>0\) such that
\(0\in K_{\varphi }\) is isolated and one of the following conditions holds
-
(i)
0 is a strict local minimizer of \(\varphi |_{V}\); or
-
(ii)
\(X=H=\) a Hilbert space and \(\varphi '\) is Lipschitz near 0,
then there exist \(\rho \in (0,r)\) and a homeomorphism \(h:X\rightarrow X\), \(h(0)=0\) such that
-
(a)
\(h(\bar{B}_{\rho }(0))\subseteq B_r(0)\);
-
(b)
\(h(u)=u\) for all \(u\in Y\cap \bar{B}_{\rho }(0)\);
-
(c)
\(\varphi (u)>0\) for all \(u\in h(V\cap \bar{B}_{\rho }(0)),\ u\ne 0\).
Proof
Suppose that hypothesis (i) holds. Then we can take \(\rho \in (0,r)\) such that
So, suppose that hypothesis (ii) holds. Let \(0<\rho _1<\rho _2<r\) be such that
Take \(\rho \in (0,\rho _1)\). Then the sets \(\bar{B}_{\rho }(0)\) and \(H\backslash B_{\rho _1}(0)\) are closed and disjoint. Consider the function \(f:H\rightarrow [0,1]\) defined by
Evidently, \(f(\cdot )\) is locally Lipschitz and \(f|_{\bar{B}_{\rho }(0)}=1\), \(f|_{H\backslash B_{\rho _1}(0)}=0\). Let \(p_V:H\rightarrow V\) the projection operator onto V and consider the map \(\xi :H\rightarrow H\) defined by
Clearly, \(\xi (\cdot )\) is Lipschitz and bounded. We consider the abstract Cauchy problem defined by
From Proposition 5.3.5 we know that problem (6.114) admits a unique global solution \(\sigma (t_0,u):\left[ 0,+\infty \right) \rightarrow H\). Consider the maps \(h, l:H\rightarrow H\) defined by
The continuous dependence of the flow on the initial condition (see Proposition 5.3.5) implies that \(h(\cdot )\) and \(l(\cdot )\) are both continuous. We have
Note that \(h(0)=0\). Also, we have:
-
(a)
If \(u\in H\backslash B_{\rho _1}(0)\), then \(\xi (u)=0\) and so \(h(u)=u\). Therefore \(h(H\backslash {\bar{B}}_{\rho _1}(0))=H\backslash \bar{B}_{\rho _1}(0)\). This mean that \(h(\bar{B}_{\rho }(0))\subseteq h(B_{\rho _1}(0))\subseteq B_{\rho _1}(0)\subseteq B_r(0)\). This proves part (a) of the lemma.
-
(b)
If \(u\in Y\), then \(\xi (u)=0\) (see (6.113)) and so \(h(u)=u\), which proves part (b) of the lemma.
-
(c)
If \(u\in V\cap \bar{B}_{\rho }(0)\), then
$$\varphi (h(u))=\varphi (u)+\int ^1_0\vartheta (t)dt,$$where \(\vartheta (t)=f(\sigma (0,u)(t))||p_V\sigma (0,u)(t)||\ ||\varphi '(\sigma (0,u)(t))||^2\). Evidently, \(\vartheta \geqslant 0\) and since \(f(u)=1,\ p_V(u)=u\), \(u\notin K_{\varphi }\), we have
$$\begin{aligned}&\vartheta (0)=||u||\ ||\nabla \varphi (u)||^2>0\\\Rightarrow & {} \varphi (h(u))>\varphi (u)\geqslant 0. \end{aligned}$$
This proves part (c) of the lemma and completes the proof. \(\square \)
Using this lemma, we obtain a precise relation between the notions of local linking and of homological local linking.
Proposition 6.6.15
If X is a Banach space, \(X=Y\oplus V\) with \(d=\mathrm{dim}\, Y<\infty \), \(\varphi \in C^1(X)\) has a local linking set at 0 (with respect to the pair (Y, V)), that is, there exists an \(r>0\) such that
\(0\in K_{\varphi }\) is isolated and one of the following conditions holds
-
(i)
0 is a strict local minimizer of \(\varphi |_V\); or
-
(ii)
\(X=H=\) a Hilbert space and \(\varphi '\) is Lipschitz near 0,
then \(\varphi \) has local (d, 1)-linking at 0.
Proof
By taking \(r>0\) even smaller if necessary, we may assume that \(K_{\varphi }\cap B_r(0)=\{0\}\). Using Lemma 6.6.14, we can find \(\rho \in (0,r)\) and a homeomorphism \(h:X\rightarrow X\) which have properties (a), (b), (c) from Lemma 6.6.14. We set
Then conditions (a) and (c) in Definition 6.6.13 follow from the above choices, the local linking property and part (c) of Lemma 6.6.14. So, we need to verify property (b) in Definition 6.6.13.
From the proof of Proposition 6.2.30, we know that
Also, from Example 6.1.34(b), we have
Since E is contractible, using Proposition 6.1.29, we see that
Axiom 4 in Definition 6.1.12 implies that
Therefore
From (6.116) and (6.117) we see that
So, property (b) in Definition 6.6.13 is satisfied. This completes the proof of the proposition. \(\square \)
Remark 6.6.16
Homological local linking in general does not imply local linking. Consider the function \(\varphi :\mathbb R^2\rightarrow \mathbb R\) defined by
Then \(\varphi \) has a local (1, 2)-linking near 0, but it does not have a local linking at 0.
Now we estimate the critical groups of \(\varphi \in C^1(X)\), when it exhibits homological linking at 0.
Theorem 6.6.17
If X is a Banach space, \(\varphi \in C^1(X),\ 0\in K_{\varphi }\) is isolated and \(\varphi \) has a local (m, n)-linking near the origin, then \(\mathrm{rank}\,C_m(\varphi , 0)\geqslant n\).
Proof
Let \(U, E_0,E\) and D be as postulated by Definition 6.6.13. From Definition 6.2.1 we have
Consider the exact chain
with \(e_*\) being the homomorphism induced by the inclusion \(e:\varphi ^0\cap U\backslash \{0\}\rightarrow \varphi ^0\cap U\). From (6.118) and the rank formula, we have
Using Definition 6.6.13, we have the following commutative diagram
Here \(l_*,\eta _*,\partial _*\) are the homomorphisms induced by the corresponding inclusion maps. Then we have
From (6.119), (6.120), (6.121) it follows that
The proof is now complete.\(\square \)
Corollary 6.6.18
If X is a Banach space, \(X=Y\oplus V\) with \(d=\mathrm{dim}\, Y<\infty \), \(\varphi \in C^1(X)\), \(\varphi \) has a local linking at \(0,\ 0\in K_{\varphi }\) is isolated and one of the conditions (i) and (ii) from Proposition 6.6.15 holds, then \(C_d(\varphi , 0)\ne 0\).
We can improve this corollary by restricting ourselves to Hilbert spaces and to \(C^2\)-functionals.
Proposition 6.6.19
If H is a Hilbert space, \(H=\bar{H}\oplus \hat{H}\) with \(d=\mathrm{dim}\,\bar{H}\), \(\varphi \in C^2(H)\), \(\varphi \) has a local linking at 0 (with respect to the pair \((\bar{H},\hat{H})\)), \(0\in K_{\varphi }\) is isolated with Morse index \(m_0\) and nullity \(\nu _0\) (that is, \(\mathrm{dim}\,\mathrm{ker}\,\varphi ''(0)=\nu _0\) and \(\varphi ''(0)\) is Fredholm), then \(C_k(\varphi , 0)=\left\{ \begin{array}{ll} \delta _{k, m_0}\mathbb Z&{}\text {if}\ d=m_0\\ \delta _{k, m_0}\mathbb Z&{}\text {if}\ d=m_0+\nu _0 \end{array}\right. \) for all \(k\in \mathbb N\).
Proof
From the shifting theorem (see Theorem 6.2.13), we have
where \(\hat{\varphi }\) is as in Proposition 6.2.9. From Corollary 6.6.18, we have
If \(d=m_0\), then 0 is a local minimizer of \(\hat{\varphi }\) and so
If \(d=m_0+\nu _0\), then 0 is a local maximizer of \(\hat{\varphi }\) and so
The proof is now complete. \(\square \)
The next result is a useful tool in the computation of critical groups at infinity.
Proposition 6.6.20
If X is a Banach space, \((t, u)\rightarrow h_t(u)\) is a function in \(C^1([0,1]\times X)\), the maps \(u\rightarrow (h_t)'(u)\) and \(t\rightarrow \partial _th_t(u)\) are both locally Lipschitz, \(h_0\) and \(h_1\) satisfy the C-condition,
with \(c_0>0,\ 1<q<p<\infty \) and there exist \(\gamma _0>0\) and \(\delta _0>0\) such that
then \(C_k(h_0,\infty )=C_k(h_1,\infty )\) for all \(k\in \mathbb N_0\).
Proof
Since by hypothesis \((t, u)\rightarrow h_t(u)\) belongs to the space \(C^1([0,1]\times X)\), it admits a pseudogradient vector field \(\hat{v}_t(u)\) (see Theorem 5.1.4). Moreover, from the construction of the pseudogradient vector field (see the proof of Theorem 5.1.4), we have
with \((t, u)\rightarrow v_t(u)\) locally Lipschitz and for all \(t\in [0,1]\), \(v_t(\cdot )\) is the pseudogradient vector field corresponding to \(h_t(\dot{)}\). So, for all \(t\in [0,1]\) and all \(u\in X\), we have
Given \(t\in [0,1]\), we consider the map \(w_t:X\rightarrow X\) defined by
Clearly, this is a well-defined vector field and \((t, u)\rightarrow w_t(u)\) is locally Lipschitz. Let \(\gamma \leqslant \gamma _0\) be such that
If no such \(\gamma \leqslant \gamma _0\) can be found, then \(C_k(h_0,\infty )=C_k(h_1,\infty )=\delta _{k, 0}\mathbb Z\) for all \(k\in \mathbb N_0\).
So, to fix things we assume that \(h^{\gamma }_0\ne 0\). Let \(y\in h^{\gamma }_0\) and consider the following abstract Cauchy problem
This Cauchy problem admits a unique local flow (see Proposition 5.3.4) denoted by \(\sigma (t, y)\). In the sequel, for notational simplicity, from \(\sigma \) we drop the initial condition y. We have
Hence for small \(t\geqslant 0\), we have
Then
\(\Rightarrow \) the flow \(\sigma (\cdot )\) is global on [0, 1].
We have that \(\sigma (t,\cdot )\) is a homeomorphism of \(h^{\gamma }_0\) onto a subset \(D_0\) of \(h^{\gamma }_1\). Reversing the time \((t\rightarrow 1-t)\) and using the corresponding flow \(\sigma _*(\cdot , v)(v\in h^{\gamma }_1)\), we have that \(h^{\gamma }_1\) is homeomorphic to a subset \(D_1\) of \(h^{\gamma }_0\).
We set
Then we have
Similarly, if
then
Recall that \(D_0\) and \(h^{\gamma }_0\) are homeomorphic. Similarly \(D_1\) and \(h^{\gamma }_1\) are homeomorphic too. These facts together with (6.125), (6.126), (6.127), (6.128) imply that
The proof is now complete.\(\square \)
From Definition 6.2.1 it is clear that if \(\varphi \in C^1(X)\) and \(u\in K_{\varphi }\) is isolated, then the critical groups of \(\varphi \) at u depend only on the values of \(\varphi \) near u. Now suppose that \(\Omega \subseteq \mathbb R^N\) is a bounded domain with a \(C^2\)-boundary \(\partial \Omega \) and let \(f:\Omega \times \mathbb R\rightarrow \mathbb R\) be a Carathéodory function, that is, for all \(x\in \mathbb R\) \(z\rightarrow f(z, x)\) is measurable and for almost all \(z\in \Omega \), \(x\rightarrow f(z, x)\) is continuous. We assume that
with \(a\in L^{\infty }(\Omega )_+\), \(2\leqslant r<2^*\) (recall that \(2^*=\left\{ \begin{array}{ll} \frac{2N}{N-2}&{}\text {if}\ N\geqslant 3,\\ +\infty &{}\text {if}\ N=1,2 \end{array}\right. \), the Sobolev critical exponent, see Definition 1.9.1). We set \(F(z, x)=\int ^x_0f(z, s)ds\) and consider the \(C^1\)-functional \(\varphi :H^1_0(\Omega )\rightarrow \mathbb R\) defined by
We recall that for \(N\geqslant 2\), the Sobolev space is not embedded into \(L^{\infty }(\Omega )\) (see Theorem 1.7.4, the Rellich–Kondrachov embedding theorem). So, if \(u_0\in K_{\varphi }\) is isolated, then a priori it seems that the critical groups of \(\varphi \) depend on values of \(f(z,\cdot )\) far away from \(u_0(z)\). We will show that in fact this is not true, establishing in an emphatic way the local character of critical groups.
First we show that without any loss of generality, we may assume that \(u_0=0\). Indeed, let
We set
Evidently, this is a Carathéodory function. Also, since \(u_0\in K_{\varphi }\), by the standard regularity theory for semilinear elliptic problems, we have \(u_0\in L^{\infty }(\Omega )\). So, it follows that \(g(z,\cdot )\) has the same polynomial growth as \(f(z,\cdot )\) (see (6.129)). We set \(G(z, x)=\int ^x_0g(z, s)ds\). Then
Therefore we can write (6.130) as follows:
Evidently, \(u=0\in K_{\hat{\varphi }}\) and it is isolated.
So, we have seen that without any loss of generality, we may assume that \(u_0=0\).
Also, let \(\delta >0\) and consider a function \(\xi \in C^1(\mathbb R)\) defined by
Let \(\psi :H^1_0(\Omega )\rightarrow \mathbb R\) be the \(C^1\)-functional defined by
In the next lemma, we compare the critical groups of \(\varphi \) and \(\psi \).
Lemma 6.6.21
If \(0\in K_{\varphi }\) is isolated, then 0 is an isolated critical point of \(\psi \) too and we have
Proof
We consider the following family of functions \(h_t(u)\) defined on \([0,1]\times H^1_0(\Omega )\)
Evidently, \(h_0(u)=\varphi (u)\) and \(h_1(u)=\psi (u)\) for all \(u\in H^1_0(\Omega )\).
We will show that \(0\in K_{h_t}\) is isolated uniformly in \(t\in [0,1]\). Arguing by contradiction, suppose we can find \(\{t_n\}_{n\geqslant 1}\subseteq [0,1]\) and \(\{u_n\}_{n\geqslant 1}\subseteq H^1_0(\Omega )\) such that
From (6.131) we have for all \(n\in \mathbb N\)
From (6.132) and the regularity theory for semilinear elliptic equations (the Calderon–Zygmund estimates), we can find \(\alpha \in (0,1)\) and \(M>0\) such that
Exploiting the compact embedding of \(C^{1,\alpha }_0(\overline{\Omega })\) into \(C^1_0(\overline{\Omega })\), from (6.131) and (6.133) we have
So, we can find \(n_0\in \mathbb N\) such that
a contradiction to our hypothesis that \(0\in K_{\varphi }\) is isolated (see (6.134)).
Having the isolation of the critical point \(u=0\) for the family \(\{h_t(\cdot )\}_{t\in [0,1]}\) we conclude that
The proof is now complete. \(\square \)
This lemma leads to the following theorem, stressing the really local character of critical groups. Its proof is immediate from Lemma 6.6.21 and the previous observations.
So, let \(f, g:\Omega \times \mathbb R\rightarrow \mathbb R\) be two Carathéodory functions satisfying
with \(a\in L^{\infty }(\Omega )\), \(2,\leqslant r<2^*\). We set
and consider the \(C^1\)-functionals \(\varphi ,\psi :H^1_0(\Omega )\rightarrow \mathbb R\) defined by
Theorem 6.6.22
If \(0\in K_{\varphi }\) is isolated and there exists a \(\delta >0\) such that
then \(0\in K_{\psi }\) is isolated too and \(C_k(\varphi , u_0)=C_k(\psi , u_0)\) for all \(k\in \mathbb N_0\).
Remark 6.6.23
Evidently, this theorem is also valid for problems other than the Dirichlet problem, such as the Neumann or Robin problems. Then \(H^1_0(\Omega )\) is replaced by \(H^1(\Omega )\).
From Palais [325] (Theorem 16), we have the following result.
Theorem 6.6.24
If \(V_1,V_2\) are two paracompact locally convex topological vector spaces, \(j:V_1\rightarrow V_2\) is a continuous, linear map, \(j(V_1)\) is dense in \(V_2\), \(W\subseteq V_2\) is open and \(U=j^{-1}(W)\), then \(\hat{j}=j|_U:U\rightarrow W\) is a homotopy equivalence.
Remark 6.6.25
Using this general result, we see that if X is a Banach space which is embedded continuously and densely into a Hilbert space H, then for any pair of open sets (D, E) in H, we have
This equality leads to the following useful result.
Theorem 6.6.26
If H is a Hilbert space, X is a Banach space which is continuously and densely embedded in H, \(\varphi \in C^2(H)\) and \(u\in K^c_{\varphi }\) is isolated, then \(C_k(\varphi ,u)=C_k(\varphi |_X, u)\) for all \(k\in \mathbb N_0\).
6.7 Existence and Multiplicity of Critical Points
In this section we use critical groups to establish the existence and multiplicity of critical points.
We start with some auxiliary results related to the so-called “Lyapunov–Schmidt reduction method”. With this method the initial infinite-dimensional problem is reduced to a finite-dimensional one which is easier to deal with. In Volume 2 we will see that this method, under some reasonable hypotheses on the data of the problem, is very effective in dealing with resonant equations.
The setting is the following. We have H, a separable Hilbert space with \(H^*\) its topological dual, and \(\left\langle \cdot ,\cdot \right\rangle \), the duality brackets for the pair \((H^*, H)\). We assume that H admits the following orthogonal direct sum decomposition
So, every \(u\in H\) admits a unique decomposition
Proposition 6.7.1
If \(\varphi \in C^1(H),\ \varphi \) is sequentially weakly lower semicontinuous and
for all \(y\in Y\), all \(v, v'\in V\) and some \(\hat{c}>0\), then there exists a continuous map \(\vartheta :Y\rightarrow V\) such that
Proof
Fix \(y\in Y\) and consider the \(C^1\)-functional \(\varphi _y:H\rightarrow \mathbb R\) defined by
Let \(i_V:V\rightarrow H\) denote the inclusion map and consider the map \(\hat{\varphi }_y:V\rightarrow H\) defined by
Evidently, \(\hat{\varphi }_y\in C^1(V, H)\) and from the chain rule we have
with \(p_{V^*}\) being the orthogonal projection of \(H^*\) onto \(V^*\) (recall that \(H^*=Y^*\oplus V^*\)). In what follows, by \(\left\langle \cdot ,\cdot \right\rangle _V\) we denote the duality brackets for the pair \((V^*, V)\). For \(v, v'\in V\) we have
For all \(v\in V\), we have
Also, note that \(\hat{\varphi }'_y:V\rightarrow V^*\) is monotone and continuous, hence by Proposition 2.6.12 \(\hat{\varphi }'_y(\cdot )\) is maximal monotone. Therefore \(\hat{\varphi }'_y\) is maximal monotone and coercive, so it is surjective (see Theorem 2.8.6). Hence, we can find \(v_0\in V\) such that
From (6.136) it is clear that \(v_0\in V\) is unique and is the unique minimizer of the strictly convex functional \(\hat{\varphi }_y=\varphi _y|_{V}\) (see (6.137)). This means that we can define the map \(\vartheta :Y\rightarrow V\) by setting \(\vartheta (y)=v_0\). Then we have
Next we show the continuity of the map \(\vartheta :Y\rightarrow V\). So, let \(y_n\rightarrow y\) in Y. For every \(n\in \mathbb N\), we have
Passing to a suitable subsequence if necessary, we may assume that
Since by hypothesis \(\varphi (\cdot )\) is sequentially weakly lower semicontinuous, we have
From (6.140) we know that
So, by the Urysohn criterion for the initial sequence \(\{\vartheta (y_n)\}_{n\geqslant 1}\subseteq V\) we have
Moreover, from (6.140) we have
The proof is now complete.\(\square \)
We set
From Proposition 6.7.1 it is clear that \(\varphi _0:Y\rightarrow \mathbb R\) is continuous. In fact, we can say more.
Proposition 6.7.2
If \(\varphi \in C^1(H),\ \varphi \) is sequentially weakly lower semicontinuous,
for all \(y\in Y\), all \(v, v'\in V\), some \(\hat{c}>0\) and \(\varphi _0:Y\rightarrow \mathbb R\) is given by (6.143), then \(\varphi _0\in C^1(Y)\).
Proof
Let \(y, h\in Y\) and \(t>0\). From (6.143) and Proposition 6.7.1, we have
Also, we have
Let \(\left\langle \cdot ,\cdot \right\rangle _Y\) denote the duality brackets for the pair \((Y^*, Y)\). From (6.144) and (6.145) we have
Similarly if \(t<0\), then
From (6.146) and (6.147) we conclude that
The proof is now complete.\(\square \)
Proposition 6.7.3
If \(\varphi \in C^1(H),\ \varphi \) is sequentially weakly lower semicontinuous,
for all \(y\in Y\), all \(v, v'\in V\), some \(\hat{c}>0\) and \(\varphi _0:Y\rightarrow \mathbb R\) is given by (6.143), then \(y\in K_{\varphi _0}\) if and only if \(y+\vartheta (y)\in K_{\varphi }\).
Proof
\(\Leftarrow :\) This is immediate from (6.135) and (6.140).
\(\Rightarrow :\) Suppose that \(y\in K_{\varphi _0}\). Then
Since \(H^*=Y^*\oplus V^*\), it follows that
The proof is now complete.\(\square \)
Remark 6.7.4
So \(y\in K_{\varphi _0}\) is isolated if and only if \(y+\vartheta (y)\in K_{\varphi }\) is isolated.
Based on this remark, one may ask what the relation is between the critical groups
The next theorem answers this equation and shows the effectiveness of the Lyapunov–Schmidt reduction method.
Theorem 6.7.5
If \(\varphi \in C^1(H),\ \varphi \) is sequentially weakly lower semicontinuous,
for all \(y\in Y\), all \(v, v'\in V\), some \(\hat{c}>0\), \(\varphi _0:Y\rightarrow \mathbb R\) is given by (6.143) and \(\hat{y}\in K_{\varphi _0}\) is isolated, then \(C_k(\varphi _0,\hat{y})=C_k(\varphi ,\hat{y}+\vartheta (\hat{y}))\) for all \(k\in \mathbb N_0.\)
Proof
Recall that \(\hat{y}+\vartheta (\hat{y})\in K_{\varphi }\) is isolated (see Proposition 6.7.3 and Remark 6.7.4).
Let \(c=\varphi _0(y)=\varphi (y+\vartheta (y))\) (see (6.143)) and
We set \(\hat{u}=\hat{y}+\vartheta (\hat{y})\in H\) and consider the maps
defined by
We have
Note that \(\eta \) is a homeomorphism and \(\eta ^{-1}(y,\vartheta (y))=y\) for all \((y,\vartheta (y))\in D\). Recall that \(v\rightarrow \varphi (y+v)\) is strictly convex (see the proof of Proposition 6.7.1). So, identifying \(u=y+v\in H\) with \(y\in Y, v\in V\) (uniquely) with the pair (y, v), we have
Therefore we can define the homotopy \(e:([0,1]\times \varphi ^c,[0,1]\times (\varphi ^c\backslash \{\hat{u}\}))\rightarrow (\varphi ^c,\varphi ^c\backslash \{\hat{u}\})\) by setting
Let \(i:(D, D\backslash \{\hat{u}\})\rightarrow (\varphi ^c,\varphi ^c\backslash \{\hat{u}\})\) be the inclusion map. Using the homotopy we can easily see that
So, the pairs \((D, D\backslash \{\hat{u}\})\) and \((\varphi ^c,\varphi ^c\backslash \{\hat{u}\})\) are homotopy equivalent. Then from Proposition 6.1.14 we have
The proof is now complete.\(\square \)
Suppose that \(\varphi \in C^1(H)\) satisfies the assumptions of the above theorem. Let \(\nabla \varphi \) denote the gradient of \(\varphi \), that is,
with \((\cdot ,\cdot )_H\) denoting the inner product of H. Suppose that
Therefore the Leray–Schauder index \(i_{LS}(\nabla \varphi ,\hat{y}+\vartheta (\hat{y}))\) (see Definition 6.2.43) can be defined. Also, since Y is finite-dimensional, the Brouwer index \(i(\varphi _0,\hat{y}, c)\) (see Definition 3.8.1) is also defined. We expect the two to be related. Indeed using Theorem 6.7.5, we have the following result.
Corollary 6.7.6
If everything is as above with \(\hat{y}\in K_{\varphi _0}\) isolated, then \(i_{LS}(\nabla \varphi ,\hat{y}+\vartheta (\hat{y}))=i(\varphi _0,\hat{y}, c)\).
Proof
Using Proposition 6.2.44 we have
The proof is now complete.\(\square \)
The next existence result is in the spirit of Proposition 6.2.42.
Proposition 6.7.7
If X is a Banach space, \(\varphi \in C^1(X)\), \(\varphi \) satisfies the C-condition, \(u\in X\), \(a,b, c\in \mathbb R\) with \(a<c<b,\ K_{\varphi }\) is finite, \(K^c_{\varphi }=\{u\}\), \(a, b\notin \varphi (K_{\varphi })\) and \(C_k(\varphi , u)\ne 0\), \(H_k(\varphi ^b,\varphi ^a)=0\) for some \(k\in \mathbb N_0\), then we can find \(u_0\in K_{\varphi }\) such that
Proof
Choose \(\epsilon >0\) small so that
and \(a<c-\epsilon<c+\epsilon <b.\)
From Proposition 6.2.16, we have
We consider the sets \(\varphi ^a\subseteq \varphi ^{c-\epsilon }\subseteq \varphi ^{c+\epsilon }\subseteq \varphi ^b\) and use Proposition 6.1.37. We obtain \(H_{k-1}(\varphi ^{c-\epsilon },\varphi ^a)\ne 0\) or \(H_{k+1}(\varphi ^b,\varphi ^{c+\epsilon })\ne 0\). Then Proposition 6.2.15 implies that we can find \(u_0\in K_{\varphi }\) such that
The proof is now complete.\(\square \)
Corollary 6.7.8
If X is a Banach space, \(\varphi \in C^1(X),\ \varphi \) satisfies the C-condition, \(K_{\varphi }\) is finite, \(K^c_{\varphi }=\{u\}\) and \(C_k(\varphi , u)\ne 0,\ C_k(\varphi ,\infty )=0\) for some \(k\in \mathbb N_0\), then we can find \(u_0\in K_{\varphi }\) such that
Proof
Since \(K_{\varphi }\) is finite, we can find \(a, b\in \mathbb R\) such that
Invoking Proposition 6.2.28(a), we have
So, we can apply Proposition 6.7.7 and find \(u_0\in K_{\varphi }\) such that
The proof is now complete.\(\square \)
Next, we present some multiplicity results.
Proposition 6.7.9
If X is a Banach space, \(\varphi \in C^1(X)\), \(\varphi \) is bounded below and satisfies the C-condition (or equivalently the PS-condition, see Proposition 5.1.14), \(u\in K_{\varphi }\) is isolated and not a global minimizer of \(\varphi \) and for some \(m\in \mathbb N_0,\ C_m(\varphi , u)\ne 0\), then \(K_{\varphi }\) has at least three elements.
Proof
Since \(\varphi \) is bounded below and satisfies the C-condition, by Proposition 5.1.8 we can find \(u_0\in K_{\varphi }\) which is a global minimizer of \(\varphi \). Since by hypothesis \(u\in K_{\varphi }\) is not a global minimizer, we must have \(u\ne u_0\) and \(\varphi (u_0)<\varphi (u)\). Suppose that \(K_{\varphi }=\{u, u_0\}\) and choose \(a, b\in \mathbb R\) such that
Then Corollary 5.3.13 implies that \(\varphi ^b\) is a strong deformation retract, while Theorem 5.3.12 (the second deformation theorem) says that \(\{u_0\}\) is a strong deformation retract of \(\varphi ^a\). So, we have
Using the log exact sequence from Proposition 6.1.29 and (6.150), (6.151), (6.152), we infer that
a contradiction to the hypothesis \(C_m(\varphi , u)\ne 0\). So, \(K_{\varphi }\) has a third element \(\hat{u}\). The proof is now complete.\(\square \)
Corollary 6.7.10
If X is a Banach space, \(\varphi \in C^1(X)\) is bounded below and satisfies the C-condition, \(\varphi \) has a local (m, n)-linking at 0 with \(m, n\in \mathbb N\) and 0 is not a global minimizer of \(\varphi \), then \(\varphi \) has at least three critical points.
Proof
From Theorem 6.6.17 we know that \(C_m(\varphi , 0)\ne 0\). So, we can use Proposition 6.7.9 and conclude that \(K_{\varphi }\) has at least three elements. \(\square \)
Proposition 6.7.11
If X is a Banach space, \(\varphi \in C^1(X)\) is bounded below and satisfies the C-condition, and \(C_m(\varphi , 0)\ne 0\) for some \(m\in \mathbb N\), then
-
(a)
\(\varphi \) has a nontrivial critical point;
-
(b)
if \(m\geqslant 2\), then \(\varphi \) has at least two nontrivial critical points.
Proof
(a) Since \(\varphi \) is bounded below and satisfies the C-condition, from Proposition 5.1.8, \(\varphi \) has a global minimizer \(u\in K_{\varphi }\). Then Proposition 6.2.3 implies
By hypothesis we have
Comparing (6.153) and (6.154) we conclude that \(u\in K_{\varphi }\) is nontrivial.
(b) From Proposition 6.2.24 we have
From (6.154), (6.155) and Proposition 6.2.42, we know that we can find \(u_0\in K_{\varphi }\) such that
Since \(1\leqslant m-1\) (recall \(m\geqslant 2\)), from (6.153) and (6.156) it follows that \(u_0\ne u\). \(\square \)
6.8 Remarks
6.1: The material on Algebraic Topology is standard and can be found in the books of Dold [148], Eilenberg and Steenrod [156], Hatcher [203], Maunder [292] and Spanier [390]. The axiomatic treatment (see Definition 6.1.12) was first introduced by Eilenberg and Steenrod [156]. The term “homology group” is due to Vietoris [410]. Singular homology was introduced by Eilenberg. Preceding that we had simplicial homology, which is the result of the work of many mathematicians, including Betti and Poincaré. The systematic introduction of group theoretic methods occurred in the 1920s through the works of Alexander, Hopf and Lefschetz, who developed simplicial homology. Cohomology theories can be axiomatized in the same way as homology theories. The formalism of category theory is helpful in this respect.
6.2: Critical groups provide a powerful tool to distinguish between critical points and to produce additional critical points of a given functional. For this reason they are important in the study of nonlinear boundary value problems. Let X be a Hausdorff topological space, \(\varphi :X\rightarrow \mathbb R\) a continuous function, \(K\subseteq X\) closed and \(c, d\in \mathbb R\), \(c\leqslant d\). As before we set
In the case when X is a Banach space and \(\varphi \) a \(C^1\)-functional, K is the critical set of \(\varphi \) (the set \(K_{\varphi }\) in the notation of Sect. 6.2).
Definition 6.8.1
If \(D\subseteq \varphi ^{-1}(c)\), the critical groups for the pair \((\varphi , D)\) are defined by
For \(u\in \varphi ^{-1}(c)\), we set
Remark 6.8.2
By excision, the critical groups depend only on the restriction of \(\varphi \) on an arbitrary neighborhood U of u. This way, in the differentiable setting we recover Definition 6.2.1, which stresses the local character of the theory.
As an easy application of Proposition 6.1.23, we get the following result.
Proposition 6.8.3
If \(D_1,D_2\subseteq \varphi ^{-1}(C)\) are disjoint closed sets then\(C_k(\varphi , D_1\cup D_2)=C_k(\varphi , D_1)\oplus C_k(\varphi , D_2)\) for all \(k\in \mathbb N_0\).
The Morse lemma (see Proposition 5.4.19) was extended to the degenerate case by Hofer [209], under the assumption that \(\varphi ''(u_0)\) is of the form of a compact perturbation of the identity. The more general form included here (see Proposition 6.2.9) is due to Mawhin and Willem [293, p. 185]. The shifting theorem (see Theorem 6.2.13) is due to Gromoll and Meyer [198]. The Morse relation (see Theorem 6.2.20) can be found in the important paper of Marino and Prodi [288] on perturbation methods in Morse theory. In the same work of Marino–Prodi we can find Lemmata 6.2.35, 6.2.37 and 6.2.38. Critical groups at infinity were introduced by Bartsch and Li [38] as the appropriate tool for the global theory of critical points. Condition \((A_{\infty })\) is a slightly more general version of the one employed by Bartsch and Li [38]. In Bartsch and Li [38] it is assumed that \(\varphi ''(u)\rightarrow 0\) as \(||u||\rightarrow \infty \). However, a careful reading of their proof reveals that it is enough to assume that \(\psi '(u)=0(||u||)\) (see condition \((A_{\infty })\)). This generalization was first used by Su and Zhao [394]. Proposition 6.2.44 is due to Rothe [361].
The next proposition gives a more convenient version of the Shifting Theorem (see Theorem 6.2.13).
Proposition 6.8.4
If H is a Hilbert space, \(\varphi \in C^2(H)\) and \(u\in K_{\varphi }\) is isolated with finite Morse index m and nullity \(\nu \), then one of the following holds:
-
(a)
\(C_k(\varphi , u)=0\) for all \(k\leqslant m\) and all \(k\geqslant m+\nu \);
-
(b)
\(C_k(\varphi ,u)=\delta _{k, m}\mathbb Z\) for all \(k\in \mathbb N_0\);
-
(c)
\(C_k(\varphi ,u)=\delta _{k, m+\nu }\mathbb Z\) for all \(k\in \mathbb N_0\).
Remark 6.8.5
In fact, the result is also true for nontrivial critical points of \(C^{2-0}\) functionals (that is, \(C^1\)-functionals \(\varphi \) whose derivative \(\varphi '(\cdot )\) is locally Lipschitz, alternatively the notation \(C^{1,1}\) is also used). This extension was proved by Li et al. [268]. In the same paper, it is also proved that Proposition 6.2.44 is in fact true for \(\varphi \in C^1(H)\).
6.3: The invariance of the critical groups with respect to small \(C^1(X)\)-perturbations (see Theorem 6.3.4) and with respect to homotopies which preserve the isolation of the critical point (see Theorem 6.3.6) are two very useful tools for computing critical groups in concrete situations. The results can be found in Chang and Ghoussoub [120] and Corvellec and Hantoute [128]. In the latter, the setting is more general (continuous functions on metric spaces using the notions of weak slope and lower critical point). The result on the continuity of critical groups with respect to the \(C^1\)-topology can also be found in Chang [118, p. 336] and in Mawhin and Willem [293, p. 196] (for \(C^2\)-functionals on Hilbert spaces). Similarly, the homotopy invariance of critical groups can also be found in Chang [118, p. 53]. Theorem 6.3.8 is due to Chang [118, p. 334].
6.4: The critical group theory can be extended to critical subsets. More precisely, we saw that an isolated critical point is replaced by a dynamically isolated critical set (see Definition 6.4.17) and for such sets we have an analogous critical group theory. In the presentation of this theory we follow Chang [118] (see also Chang and Ghoussoub [120]), who developed an extension of the Gromoll–Meyer theory (see Gromoll and Meyer [198]). This extended theory is also related to the Conley index theory of isolated invariant sets for gradient flows.
6.5: Recall that the notion of a critical point of mountain pass type (see Definition 6.5.6) is due to Hofer [210, 211]. Theorem 6.5.8 is usually proved using the second deformation theorem (see Theorem 5.3.12) and arguments based on the existence of neighborhoods which are stable with respect to the pseudogradient flow. Here, we follow a different approach based on Lemma 6.2.35. This allows us to slightly weaken the hypotheses in Theorem 6.5.8 and assume that the critical value \(\varphi (u_0)\) is isolated in \(\varphi (K_{\varphi })\). Theorem 6.5.11 can also be found in Chang [118, p. 91] and in Mawhin and Willem [293, p. 195], under a little more restrictive conditions (see also Bartsch [37]).
6.6: The notion of homological linking (see Definition 6.6.1) goes back to the work of Liu [276], who also proved the nontriviality of the first critical group \(C_1(\varphi , u_0)\) for a critical point \(u_0\in K_{\varphi }\) produced by an application of the mountain pass theorem (see also Corollary 6.6.9). The notion of local (m, n)-linking (see Definition 6.6.13) is due to Perera [335], who also proved Theorem 6.6.17. Corollary 6.6.18 is due to Liu [276]. Proposition 6.6.20 can be found in Papageorgiou and Rădulescu [330]. Theorem 6.6.22, stressing the local character of critical groups, is essentially due to Degiovanni et al. [141], who employed an approach using the truncation function \(\xi (\cdot )\) (see Lemma 6.6.21). Theorem 6.6.26 can also be found in Chang [118, p. 14] and in Bartsch [36].
6.7: The “reduction method” for elliptic problems was developed by Amann [12, 13] and Castro and Lazer [109]. Theorem 6.7.5 is due to Liu and Li [278] and Liu [276]. Multiplicity results for critical points using critical groups can also be found in Motreanu et al. [309, 310] and in Perera et al. [336].
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Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D. (2019). Morse Theory and Critical Groups. In: Nonlinear Analysis - Theory and Methods. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-03430-6_6
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