Variations

Abstract

In this chapter we discuss variations of functionals. The idea is the following: Let $\mathscr {F}:\mathrm {W}^{1,p}_g(\varOmega ;\mathbb {R}^m) \rightarrow \mathbb {R}$ be a functional with minimizer $u_* \in \mathrm {W}^{1,p}_g(\varOmega ;\mathbb {R}^m)$. Take a path $t \mapsto u_t \in \mathrm {W}^{1,p}_g(\varOmega ;\mathbb {R}^m)$ ($t \in \mathbb {R}$) with $u_0 = u_*$ and consider the behavior of the map

$$ t \mapsto \mathscr {F}[u_t] $$

around $t = 0$. If $t \mapsto \mathscr {F}[u_t]$ is differentiable at $t = 0$, then its derivative at $t = 0$ must vanish because of the minimization property. This is analogous to the elementary fact that if $g \in \mathrm {C}^1((0,T))$ takes its minimum at a point $t_* \in (0,T)$, then $g'(t_*) = 0$.

Appendices

Notes and Historical Remarks

Difference quotients were already considered by Newton. Their application to regularity theory is due to work by Nirenberg in the 1940s and 1950s. Many of the fundamental results of regularity theory (albeit in a non-variational context) can be found in [136]. The books by Giusti [137] and Giaquinta–Martinazzi [133] contain theory relevant for variational questions, whereas [177] treats many questions of “fine regularity” (e.g. pointwise properties of solutions). A recent, very accessible introduction to regularity theory for PDEs is [36], also see the survey [188], which focuses on the calculus of variations. A nice framework for Schauder estimates is [244].

The Fundamental Lemma 3.10 of the calculus of variations is due to Paul Du Bois-Reymond and is sometimes named after him.

Noether’s theorem has many ramifications and can be put into a very general form in Hamiltonian systems and Lie group theory. The idea is to study groups of symmetries and their actions. For an introduction to this diverse field, see [218] and also [278]. Example 3.26 about the monotonicity formula is from [111]. For more on Lagrange multipliers and Noether’s theorem, see [131, 132].

Subdifferentials were introduced in the general work of Jean-Jacques Moreau and R. Tyrrell Rockafellar on convex analysis. There are also extended subdifferentials for non-convex functions; see the monographs [233] and [192, 193] for more on this.

Problems

3.1.

Let $\varOmega \subset \mathbb {R}^d$ be a bounded Lipschitz domain.

• (i) Compute the Euler–Lagrange equation, in weak form, for a minimizer $u \in \mathrm {W}^{1,2}(\varOmega )$ of the functional

  $$ \mathscr {F}[u] := \int _\varOmega \frac{1}{2} \nabla u(x) S(x) \nabla u(x)^T - g(x) u(x) \;\mathrm{d}x, $$

  where $S :\varOmega \rightarrow \mathbb {R}^{d \times d}$, $g :\varOmega \rightarrow \mathbb {R}$ are continuous and $S(x) = S(x)^T$ for all $x \in \varOmega$.

• (ii) Assume now that additionally S(x) is continuously differentiable in x and that $u \in \mathrm {W}^{2,2}(\varOmega )$ is a minimizer of $\mathscr {F}$ as above. State the strong Euler–Lagrange equation for u and prove that it follows from the weak version.

3.2.

Let $\varOmega \subset \mathbb {R}^2$ and let $u \in \mathrm {W}^{1,2}(\varOmega ;\mathbb {R}^2)$. In the three cases

• (i) $f(A) = A^i_j$ for some $i, j \in \{1,2\}$,

• (ii) $f(A) = \det A$,

• (iii) $f(A) = ({{\mathrm{cof}}}A)^i_j$ for some $i, j \in \{1,2\}$

show through a direct calculation that

$$\begin{aligned} - {{\mathrm{div}}}[ \mathrm {D}_A f(\nabla u) ] = 0 \end{aligned}$$

(3.27)

for all $u \in \mathrm {C}^2(\varOmega ;\mathbb {R}^2)$. This shows that these f are null-Lagrangians , i.e., the Euler–Lagrange equation holds for all u.

3.3.

Show that also for $d \ge 3$ the above Euler–Lagrange equation (3.27) continues to hold for all $(r \times r)$-minors $f(A) = M(A)$, that is, M(A) is the determinant of a selection of r rows and r columns of A. Hint: You can assume that you select the first r rows and columns, thus considering only the principal minors.

3.4.

Prove Theorem 3.14, namely that for functionals of the form

$$ \mathscr {F}[u] := \int _\varOmega f(\nabla u(x)) - h(x) \cdot u(x) \;\mathrm{d}x, \qquad u \in \mathrm {W}^{1,2}(\varOmega ;\mathbb {R}^m), $$

where f satisfies the same assumptions as in Theorem 3.11 and $h \in \mathrm {L}^2(\varOmega ;\mathbb {R}^m)$, the minimizer $u_*$ has $\mathrm {W}^{2,2}_\mathrm {loc}$-regularity. Show furthermore that if f is quadratic and h is smooth, then $u_* \in \mathrm {C}^\infty (\varOmega ;\mathbb {R}^m)$.

3.5.

Prove an analogue of Theorem 3.11 for functionals of the form

$$ \mathscr {F}[u] := \int _\varOmega f(\nabla u(x)) - H(u(x)) \;\mathrm{d}x, \qquad u \in \mathrm {W}^{1,2}(\varOmega ), $$

where f satisfies the same assumptions as in Theorem 3.11 and $H :\mathbb {R}^m \rightarrow \mathbb {R}$ is continuously differentiable with $|\mathrm {D}H(v)| \le C(1+|v|)$ for some $C > 0$ and all $v \in \mathbb {R}^m$. Is it possible to also allow the weaker growth bound $|\mathrm {D}H(v)| \le C(1+|v|^r)$ for some $r > 1$?

3.6.

Consider the minimization problem for the problem of linearized elasticity on $\varOmega \subset \mathbb {R}^3$,

$$ \left\{ \begin{aligned}&\text {Minimize} \quad \mathscr {F}[u] := \int _\varOmega \mu |\mathscr {E}u(x)|^2 + \frac{1}{2} \Bigl ( \kappa - \frac{2}{3}\mu \Bigr ) |{{\mathrm{tr}}}\mathscr {E}u(x)|^2 - b(x) \cdot u(x) \;\mathrm{d}x\\&\text {over all} \quad \quad u \in \mathrm {W}^{1,2}(\varOmega ;\mathbb {R}^3)~\text {with}~u|_{\partial \varOmega } = g, \end{aligned} \right. $$

where $\mu , \kappa > 0$ are such that $\kappa - \frac{2}{3}\mu \ge 0$, $f \in \mathrm {L}^\infty (\varOmega ;\mathbb {R}^3)$, $g \in \mathrm {W}^{1/2,2}(\partial \varOmega ;\mathbb {R}^3)$, and $\mathscr {E}u(x) := (\nabla u(x) + \nabla u(x)^T)/2$. Prove that the Euler–Lagrange equation (satisfied by a minimizer in a weak sense) is

$$ \left\{ \begin{aligned} - {{\mathrm{div}}}\biggl [ 2 \mu \, \mathscr {E}u + \Bigl ( \kappa - \frac{2}{3}\mu \Bigr ) ({{\mathrm{tr}}}\mathscr {E}u)I \biggr ]&= b \quad \text {in}~\varOmega ,\\ u&= g \quad \text {on}~\partial \varOmega . \end{aligned} \right. $$

3.7.

In the situation of the previous problem, assume $\kappa - \frac{2}{3}\mu = 0$, $b = 0$, and that $u \in (\mathrm {W}^{1,2} \cap \mathrm {W}^{2,2}_\mathrm {loc})(\varOmega ;\mathbb {R}^3)$ is a minimizer of $\mathscr {F}$ as above. Show, using a suitable Noether symmetry, that for all skew-symmetric $W \in \mathbb {R}^{3 \times 3}$ ($W^T = -W$) it holds that

$$ {{\mathrm{div}}}[ x^T W^T \mathscr {E}u(x) ] = 0 \qquad \text {for a.e.}\ x \in \varOmega . $$

3.8.

Set for a skew-symmetric $W \in \mathbb {R}^{d \times d}$

$$ x_\tau = g(x,\tau ) := \exp (\tau W) x, \qquad u_\tau = H(x,\tau ) := u( \exp (\tau W) x ), \qquad \tau \in \mathbb {R}. $$

Show that the Dirichlet functional is invariant under the rotational transformation defined by (g, H) and conclude that any minimizer $u_* \in (\mathrm {W}^{1,2} \cap \mathrm {W}^{2,2}_\mathrm {loc})(\varOmega ;\mathbb {R}^m)$ of the Dirichlet functional (for given boundary values) satisfies the conservation law

$$ {{\mathrm{div}}}\bigl [ x^T W |\nabla u_*(x)|^2 \bigr ] = 0. $$

3.9.

In the situation of Exercise 2.2, derive the weak Euler–Lagrange equation. Hint: Think about the class of “test variations” $\psi$ that you need to allow.

3.10.

In the situation of Example 3.22, show that $E > 0$ is the smallest eigenvalue of the operator $\varPsi \mapsto [\frac{-\hbar ^2}{2\mu } \Delta + V(x)]\varPsi$.