Abstract
We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev–Slobodeckiĭ norm. We compare it to the fractional Sobolev space obtained by the K-method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.
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1 Introduction
1.1 Motivations
In the recent years, there has been a great surge of interest toward Sobolev spaces of fractional order. This is a very classical topic, essentially initiated by the Russian school in the 1950s of the last century, with the main contributions given by Besov, Lizorkin, Nikol’skiĭ, Slobodeckiĭ and their collaborators. Nowadays, we have a lot of monographies at our disposal on the subject. We just mention the books by Adams [1, 2], by Nikol’skiĭ [25] and by Triebel [30,31,32]. We also refer the reader to [31, Chapter 1] for an historical introduction to the subject.
The reason for this revival lies in the fact that fractional Sobolev spaces seem to play a fundamental role in the study and description of a vast amount of phenomena, involving nonlocal effects. Phenomena of this type have a wide range of applications; we refer to [10] for an overview.
There are many ways to introduce fractional derivatives and, consequently, Sobolev spaces of fractional order. Without any attempt of completeness, let us mention the two approaches which are of interest for our purposes:
-
a concrete approach, based on the introduction of explicit norms, which are modeled on the case of Hölder spaces. For example, by using the heuristic
$$\begin{aligned} \delta _h^s u(x):=\frac{u(x+h)-u(x)}{|h|^s}\sim \hbox { ``derivative of order }{\textit{s}}\hbox {''},\quad \hbox { for }x,h\in {\mathbb {R}}^N, \end{aligned}$$a possible choice of norm is
$$\begin{aligned} \left( \int \left\| \delta _h^s u\right\| _{L^p}^p\,\frac{\mathrm{d}h}{|h|^N}\right) ^\frac{1}{p}, \end{aligned}$$and more generally
$$\begin{aligned} \left( \int \left\| \delta _h^s u\right\| _{L^p}^q\,\frac{\mathrm{d}h}{|h|^N}\right) ^\frac{1}{q},\quad \hbox { for } 1\le q\le \infty . \end{aligned}$$Observe that the integral contains the singular kernel \(|h|^{-N}\); thus, functions for which the norm above is finite must be better than just merely s-Hölder regular, in an averaged sense;
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an abstract approach, based on the so-called interpolation methods. The foundations of these methods were established at the beginning of the 1960s of the last century, by Calderón, Gagliardo, Krejn, Lions and Petree, among others. A comprehensive treatment of this approach can be found for instance in the books [3, 4, 29] and references therein.
In a nutshell, the idea is to define a scale of “intermediate spaces” between \(L^p\) and the standard Sobolev space \(W^{1,p}\), by means of a general abstract construction. The main advantage of this second approach is that many of the properties of the spaces constructed in this way can be extrapolated in a direct way from those of the two “endpoint” spaces \(L^p\) and \(W^{1,p}\).
As mentioned above, actually other approaches are possible: a possibility is to use the Fourier transform. Another particularly elegant approach consists in taking the convolution with a suitable kernel (for example, heat or Poisson kernels are typical choices) and looking at the rate of blowup of selected \(L^p\) norms with respect to the convolution parameter. However, we will not consider these constructions in the present paper; we refer the reader to [31] for a wide list of definitions of this type.
In spite of the explosion of literature on Calculus of Variations settled in fractional Sobolev spaces of the last years, the abstract approach based on interpolation seems to have been completely neglected or, at least, overlooked. For example, the well-known survey paper [14], which eventually became a standard reference on the field, does not even mention interpolation techniques.
1.2 Aims
The main scope of this paper is to revitalize some interest toward interpolation theory in the context of fractional Sobolev spaces. In doing this, we will resist the temptation of any unnecessary generalization. Rather, we will focus on a particular, yet meaningful, question which can be resumed as follows:
We can already anticipate the conclusions of the paper and say that this is not always true. Let us now try to enter more in the details of the present paper.
Our concerns involve the so-called homogeneous fractional Sobolev–Slobodeckiĭ spaces\({\mathcal {D}}^{s,p}_0(\varOmega )\). Given an open set \(\varOmega \subset {\mathbb {R}}^N\), an exponent \(1\le p <\infty \) and a parameter \(0<s<1\), this space is defined as the completion of \(C_0^\infty (\varOmega )\) with respect to the norm
Such a space is the natural fractional counterpart of the homogeneous Sobolev space \({\mathcal {D}}^{1,p}_0(\varOmega )\), defined as the completion of \(C^\infty _0(\varOmega )\) with respect to the norm
The space \({\mathcal {D}}^{1,p}_0(\varOmega )\) has been first studied by Deny and Lions in [13], among others. We recall that \({\mathcal {D}}^{1,p}_0(\varOmega )\) is a natural setting for studying variational problems of the type
supplemented with Dirichlet boundary conditions, in the absence of regularity assumptions on the boundary \(\partial \varOmega \). In the same way, the space \({\mathcal {D}}^{s,p}_0(\varOmega )\) is the natural framework for studying minimization problems containing functionals of the type
in the presence of nonlocal Dirichlet boundary conditions, i.e., the values of u are prescribed on the whole complement \({\mathbb {R}}^N{{\setminus }}\varOmega \). Observe that even if this kind of boundary conditions may look weird, these are the correct ones when dealing with energies (1.1), which take into account interactions “from infinity.”
The connection between the two spaces \({\mathcal {D}}^{1,p}_0(\varOmega )\) and \({\mathcal {D}}^{s,p}_0(\varOmega )\) is better appreciated by recalling that for \(u\in C^\infty _0(\varOmega )\), we have (see [5] and [26, Corollary 1.3])
with
On the other hand, as \(s\searrow 0\) we have (see [24, Theorem 3])
with
and \(\omega _N\) is the volume of the N-dimensional unit ball. These two results reflect the “interpolative” nature of the space \({\mathcal {D}}^{s,p}_0(\varOmega )\), which will be, however, discussed in more detail in the sequel.
Indeed, one of our goals is to determine whether \({\mathcal {D}}^{s,p}_0(\varOmega )\) coincides or not with the real interpolation space\({\mathcal {X}}^{s,p}_0(\varOmega )\) defined as the completion of \(C_0^\infty (\varOmega )\) with respect to the norm
Here \(K(t,\cdot ,L^p(\varOmega ),{\mathcal {D}}^{1,p}_0(\varOmega ))\) is the K-functional associated with the spaces \(L^p(\varOmega )\) and \({\mathcal {D}}^{1,p}_0(\varOmega )\), see Sect. 3 for more details.
In particular, we will be focused on obtaining double-sided norm inequalities leading to answer our initial question, i.e., estimates of the form
Moreover, we compute carefully the dependence on the parameter s of the constant C. Indeed, we will see that C can be taken independent of s.
1.3 Results
We now list the main achievements of our discussion:
-
1.
the space \({\mathcal {D}}^{s,p}_0(\varOmega )\) is always larger than \({\mathcal {X}}^{s,p}_0(\varOmega )\) (see Proposition 4.1) and they do not coincide for general open sets, as we exhibit with an explicit example (see Example 4.4);
-
2.
they actually coincide on a large class of domains, i.e., bounded convex sets (Theorem 4.7), convex cones (Corollary 4.8), Lipschitz sets (Theorem 4.10);
-
3.
the Poincaré constants for the embeddings
$$\begin{aligned} {\mathcal {D}}^{s,p}_0(\varOmega )\hookrightarrow L^p(\varOmega ) \quad \hbox {and}\quad {\mathcal {D}}^{1,p}_0(\varOmega )\hookrightarrow L^p(\varOmega ), \end{aligned}$$are equivalent for the classes of sets at point 2 (Theorem 6.1). More precisely, by setting
$$\begin{aligned} \lambda ^s_p(\varOmega )=\inf _{u\in C_0^\infty (\varOmega )}\Big \{ [u]^p_{W^{s,p}(\mathbb {R}^N)}\ :\ \Vert u\Vert _{L^p(\varOmega )}=1\Big \},\quad 0<s<1, \end{aligned}$$and
$$\begin{aligned} \lambda ^1_p(\varOmega )=\inf _{u\in C_0^\infty (\varOmega )}\left\{ \int _\varOmega |\nabla u|^p\,\mathrm{d}x\, :\, \Vert u\Vert _{L^p(\varOmega )}=1\right\} , \end{aligned}$$we have
$$\begin{aligned} \frac{1}{C}\,\Big (\lambda ^1_p(\varOmega )\Big )^s\le s\,(1-s)\,\lambda ^s_p(\varOmega )\le C\, \Big (\lambda ^1_p(\varOmega )\Big )^s. \end{aligned}$$Moreover, on convex sets the constant \(C>0\) entering in the relevant estimate is universal, i.e., it depends onNandponly. On the other hand, we show that this equivalence fails if we drop any kind of regularity assumptions on the sets (see Remark 6.3).
As a byproduct of our discussion, we also highlight some weird and unnatural behaviors of the interpolation space \({\mathcal {X}}^{s,p}_0(\varOmega )\):
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the “extension by zero” operator \({\mathcal {X}}^{s,p}_0(\varOmega )\hookrightarrow {\mathcal {X}}_0^{s,p}({\mathbb {R}}^N)\) is not an isometry for general open sets (see Remark 4.6) and the two norms
$$\begin{aligned} \Vert \,\cdot \,\Vert _{{\mathcal {X}}^{s,p}_0(\varOmega )}\quad \hbox {and}\quad \Vert \,\cdot \,\Vert _{{\mathcal {X}}^{s,p}_0({\mathbb {R}}^N)}, \end{aligned}$$may not be equivalent on \(C^\infty _0(\varOmega )\). This is in contrast with what happens for the spaces \(L^p(\varOmega )\), \({\mathcal {D}}^{1,p}_0(\varOmega )\) and \({\mathcal {D}}^{s,p}_0(\varOmega )\);
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the sharp Poincaré interpolation constant
$$\begin{aligned} \varLambda ^s_p(\varOmega )=\inf _{u\in C^\infty _0(\varOmega )} \Big \{ \Vert u\Vert _{{\mathcal {X}}^{s,p}_0(\varOmega )}^p\ :\ \Vert u\Vert _{L^p(\varOmega )}=1\Big \},\quad 0<s<1 \end{aligned}$$is sensitive to removing sets with zero capacity. In other words, if we remove a compact set \(E\Subset \varOmega \) having zero capacity in the sense of \({\mathcal {X}}^{s,p}_0(\varOmega )\), it may happen that (see Lemma 5.4)
$$\begin{aligned} \varLambda ^s_p(\varOmega {\setminus } E)>\varLambda ^s_p(\varOmega ). \end{aligned}$$Again, this is in contrast with the case of \({\mathcal {D}}^{1,p}_0(\varOmega )\) and \({\mathcal {D}}^{s,p}_0(\varOmega )\).
Remark 1.1
As recalled at the beginning, nowadays there is a huge literature on Sobolev spaces of fractional order. Nevertheless, to the best of our knowledge, a detailed discussion on the space \({\mathcal {D}}^{s,p}_0(\varOmega )\) in connection with interpolation theory seems to be missing. For this reason, we believe that our discussion is of independent interest.
We also point out that for Sobolev spaces of functions not necessarily vanishing at the boundary, there is a very nice paper [11] by Chandler-Wilde, Hewett and Moiola comparing “concrete” constructions with the interpolation one.
1.4 Plan of the paper
In Sect. 2 we present the relevant Sobolev spaces, constructed with the concrete approach based on the so-called Sobolev–Slobodeckiĭ norms. Then in Sect. 3 we introduce the homogeneous interpolation space we want to work with. Essentially, no previous knowledge of interpolation theory is necessary.
The comparison between the concrete space and the interpolation one is contained in Sect. 4. This in turn is divided in three subsections, each one dealing with a different class of open sets. We point out here that we preferred to treat convex sets separately from Lipschitz sets, for two reasons: the first one is that for convex sets the comparison between the two spaces can be done “by hands,” without using any extension theorem. This in turn permits to have a better control on the relevant constants entering in the estimates. The second one is that in proving the result for Lipschitz sets, we actually use the result for convex sets.
In order to complement the comparison between the two spaces, in Sect. 5 we compare the two relevant notions of capacity, naturally associated with the norms of these spaces. Finally, Sect. 6 compares the Poincaré constants.
The paper ends with three appendices: the first one contains the construction of a counterexample used throughout the whole paper; the second one proves a version of the one-dimensional Hardy inequality; and the last one contains a geometric expedient result for convex sets.
2 Preliminaries
2.1 Basic notation
In what follows, we will always denote by N the dimension of the ambient space. For an open set \(\varOmega \subset {\mathbb {R}}^N\), we indicate by \(|\varOmega |\) its N-dimensional Lebesgue measure. The symbol \({\mathcal {H}}^k\) will stand for the k-dimensional Hausdorff measure. Finally, we set
and
2.2 Sobolev spaces
For \(1\le p<\infty \) and an open set \(\varOmega \subset {\mathbb {R}}^N\), we use the classical definition
This is a Banach space endowed with the norm
We also denote by \({\mathcal {D}}^{1,p}_0(\varOmega )\) the homogeneous Sobolev space, defined as the completion of \(C^\infty _0(\varOmega )\) with respect to the norm
If the open set \(\varOmega \subset {\mathbb {R}}^N\) supports the classical Poincaré inequality
then \({\mathcal {D}}^{1,p}_0(\varOmega )\) is indeed a functional space and it coincides with the closure in \(W^{1,p}(\varOmega )\) of \(C^\infty _0(\varOmega )\). We will set
It occurs \(\lambda ^1_p(\varOmega )=0\) whenever \(\varOmega \) does not support such a Poincaré inequality.
Remark 2.1
We remark that one could also consider the space
It is easy to see that \({\mathcal {D}}^{1,p}_0(\varOmega )\subset {W}^{1,p}_0(\varOmega )\), whenever \({\mathcal {D}}^{1,p}_0(\varOmega )\hookrightarrow L^p(\varOmega )\). If in addition \(\partial \varOmega \) is continuous, then both spaces are known to coincide, thanks to the density of \(C^\infty _0(\varOmega )\) in \(W^{1,p}_0(\varOmega )\), see [20, Theorem 1.4.2.2].
2.3 A homogeneous Sobolev–Slobodeckiĭ space
Given \(0<s< 1\) and \(1\le p <\infty \), the fractional Sobolev space \(W^{s,p}({\mathbb {R}}^N)\) is defined as
where the Sobolev–Slobodeckiĭ seminorm\([\,\cdot \,]_{W^{s,p}({\mathbb {R}}^N)}\) is defined as
This is a Banach space endowed with the norm
In what follows, we need to consider nonlocal homogeneous Dirichlet boundary conditions, outside an open set \(\varOmega \subset {\mathbb {R}}^N\). In this setting, it is customary to consider the homogeneous Sobolev–Slobodeckiĭ space\({\mathcal {D}}^{s,p}_0(\varOmega )\). The latter is defined as the completion of \(C^\infty _0(\varOmega )\) with respect to the norm
Observe that the latter is indeed a norm on \(C^\infty _0(\varOmega )\). Whenever the open set \(\varOmega \subset {\mathbb {R}}^N\) admits the following Poincaré inequality
we get that \({\mathcal {D}}^{s,p}_0(\varOmega )\) is a functional space continuously embedded in \(L^p(\varOmega )\). In this case, it coincides with the closure in \(W^{s,p}({\mathbb {R}}^N)\) of \(C^\infty _0(\varOmega )\). We endow the space \({\mathcal {D}}^{s,p}_0(\varOmega )\) with the norm
We also define
i.e., this is the sharp constant in the relevant Poincaré inequality. Some embedding properties of the space \({\mathcal {D}}^{s,p}_0(\varOmega )\) are investigated in [18].
Remark 2.2
As in the local case, one could also consider the space
It is easy to see that \({\mathcal {D}}^{s,p}_0(\varOmega )\subset {W}^{s,p}_0(\varOmega )\), whenever \({\mathcal {D}}^{s,p}_0(\varOmega )\hookrightarrow L^p(\varOmega )\). As before, if \(\partial \varOmega \) is continuous, then both spaces are known to coincide, again thanks to the density of \(C^\infty _0(\varOmega )\) in \(W^{s,p}_0(\varOmega )\), see [20, Theorem 1.4.2.2].
2.4 Another space of functions vanishing at the boundary
Another natural fractional Sobolev space of functions “vanishing at the boundary” is given by the completion of \(C^\infty _0(\varOmega )\) with respect to the localized norm
We will denote this space by \(\mathring{D}^{s,p}(\varOmega )\) and endow it with the norm \([\Vert u\Vert _{\mathring{D}^{s,p}(\varOmega )}:=[u]_{W^{s,p}(\varOmega )}]\). We recall the following
Lemma 2.3
Let \(1<p<\infty \) and \(0<s<1\). For every \(\varOmega \subset {\mathbb {R}}^N\) open bounded Lipschitz set, we have:
-
if \(s\,p>1\), then
$$\begin{aligned} {\mathcal {D}}^{s,p}_0(\varOmega )=\mathring{D}^{s,p}(\varOmega ); \end{aligned}$$ -
if \(s\,p\le 1\), then there exists a sequence \(\{u_n\}_{n\in {\mathbb {N}}}\subset C^\infty _0(\varOmega )\) such that
$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{\Vert u_n\Vert _{\mathring{D}^{s,p}(\varOmega )}}{\Vert u_n\Vert _{{\mathcal {D}}^{s,p}_0(\varOmega )}}=0. \end{aligned}$$
Proof
The proof of the first fact is contained in [7, Proposition B.1].
As for the case \(s\,p\le 1\), in [15, Section 2] Dyda constructed a sequence \(\{u_n\}_{n\in {\mathbb {N}}}\subset C^\infty _0(\varOmega )\) such that
By observing that for such a sequence we have
we get the desired conclusion, by observing that \(\lambda ^s_p(\varOmega )>0\) for an open bounded set, thanks to [8, Corollary 5.2]. \(\square \)
Remark 2.4
Clearly, we always have
As observed in [16], the reverse inequality
is equivalent to the validity of the Hardy-type inequality
A necessary and sufficient condition for this to happen is proved in [16, Proposition 2]. We also observe that the failure of (2.1) implies that in general the “extension by zero” operator
is not continuous. We refer to [16] for a detailed discussion of this issue.
Remark 2.5
The space \(\mathring{D}^{s,p}(\varOmega )\) is quite problematic in general, especially in the case \(s\,p\le 1\) where it may fail to be a functional space. A more robust variant of this space is
By definition, this is automatically a functional space, continuously contained in \(W^{s,p}(\varOmega )\). It is a classical fact that if \(\varOmega \) is a bounded open set with smooth boundary, then
see [32, Theorem 3.4.3]. Moreover, we also have
see, for example, [7, Proposition B.1].
3 An interpolation space
Let \(\varOmega \subset {\mathbb {R}}^N\) be an open set. If \(X(\varOmega )\) and \(Y(\varOmega )\) are two normed vector spaces containing \(C^\infty _0(\varOmega )\) as a dense subspace, we define for every \(t>0\) and \(u\in C^\infty _0(\varOmega )\) the K-functional
We are interested in the following specific case: let us take \(0<s<1\) and \(1<p<\infty \), we choose
Then we use the notation
It is standard to see that this is a norm on \(C^\infty _0(\varOmega )\), see [4, Section 3.1]. We will indicate by \({\mathcal {X}}^{s,p}_0(\varOmega )\) the completion of \(C^\infty _0(\varOmega )\) with respect to this norm.
The first result is the Poincaré inequality for the interpolation space \({\mathcal {X}}^{s,p}_0(\varOmega )\). The main focus is on the explicit dependence of the constant on the local Poincaré constant \(\lambda ^1_{p}\).
Lemma 3.1
Let \(1<p<\infty \) and \(0<s<1\). Let \(\varOmega \subset {\mathbb {R}}^N\) be an open set. Then for every \(u\in C^\infty _0(\varOmega )\) we have
Proof
We proceed in two stages: we first prove that
and then we show that the last integral is estimated from the above by the norm \({\mathcal {X}}^{s,p}_0(\varOmega )\).
First stage Let us take \(u\in C^\infty _0(\varOmega )\), for every \(t\ge 1\) and \(v\in C^\infty _0(\varOmega )\)
By taking the infimum, we thus get
By integrating with respect to the singular measure \(\mathrm{d}t/t\), we then get
We now pick \(0<t<1\), by triangle inequality we get for every \(v\in C^\infty _0(\varOmega )\)
By taking the infimum over \(v\in C^\infty _0(\varOmega )\), we obtain for \(u\in C^\infty _0(\varOmega )\) and \(0<t<1\)
By integrating again, we get this time
By summing up (3.3) and (3.4), we get the estimate
Second stage Given \(u\in C^\infty _0(\varOmega )\), we take \(v\in C^\infty _0(\varOmega )\). We can suppose that \(\lambda ^1_p(\varOmega )>0\); otherwise, (3.2) trivially holds. By definition of \(\lambda ^1_{p}(\varOmega )\), we have that
If we recall the definition (3.1) of the K-functional, we get
and by taking infimum over \(v\in C^\infty _0(\varOmega )\) and multiplying by \(t^{-s\,p}\), we get
We integrate over \(t>0\), by performing the change of variable \(\tau =t/(\lambda _{p}^1(\varOmega ))^\frac{1}{p}\) we get
By using this in (3.5), we prove the desired inequality (3.2). \(\square \)
We will set
i.e., this is the sharp constant in the relevant Poincaré inequality. As a consequence of (3.2), we obtain
Proposition 3.2
(Interpolation inequality) Let \(1<p<\infty \) and \(0<s<1\). Let \(\varOmega \subset {\mathbb {R}}^N\) be an open set. For every \(u\in C^\infty _0(\varOmega )\) we have
In particular, we also obtain
Proof
We can assume that \(u\not \equiv 0\); otherwise, there is nothing prove. In the definition of the K-functional \(K(t,u,L^p(\varOmega ),{\mathcal {D}}^{1,p}_0(\varOmega ))\), we take \(v=\tau \,u\) for \(\tau >0\); thus, we obtain
By raising to the power p and integrating for \(t>0\), we get
We thus get the desired conclusion (3.7). The estimate (3.8) easily follows from the definition of Poincaré constant. \(\square \)
From (3.6) and (3.8), we get, in particular, the following
Corollary 3.3
(Equivalence of Poincaré constants) Let \(1<p<\infty \) and \(0<s<1\). For every \(\varOmega \subset {\mathbb {R}}^N\) open set we have
In particular, there holds
Remark 3.4
(Extensions by zero in \({\mathcal {X}}^{s,p}_0\)) We observe that by interpolating the “extension by zero” operators
which are both continuous, one obtains the same result for the interpolating spaces. In other words, we have
This can be also seen directly: it is sufficient to observe that \(C^\infty _0(\varOmega )\subset C^\infty _0({\mathbb {R}}^N)\); thus, we immediately get
since in the K-functional on the left-hand side the infimum is performed on a larger class. By integrating, we get the conclusion.
However, differently from the case of \({\mathcal {D}}^{1,p}_0(\varOmega )\), \(L^p(\varOmega )\) and \({\mathcal {D}}^{s,p}_0(\varOmega )\), in general for \(u\in C^\infty _0(\varOmega )\) we have
In other words, even if \(u\equiv 0\) outside \(\varOmega \), passing from \(\varOmega \) to \({\mathbb {R}}^N\) has an impact on the interpolation norm.
Actually, if \(\varOmega \) has not smooth boundary, the situation can be much worse than this. We refer to Remark 4.6.
4 Interpolation versus Sobolev–Slobodeckiĭ
4.1 General sets
We want to compare the norms of \({\mathcal {D}}^{s,p}_0(\varOmega )\) and \({\mathcal {X}}^{s,p}_0(\varOmega )\). We start with the simplest estimate, which is valid for every open set.
Proposition 4.1
(Comparison of norms I) Let \(1<p<\infty \) and \(0<s<1\). Let \(\varOmega \subset {\mathbb {R}}^N\) be an open set, then for every \(u\in C^\infty _0(\varOmega )\) we have
In particular, we have the continuous inclusion \({\mathcal {X}}^{s,p}_0(\varOmega )\subset {\mathcal {D}}^{s,p}_0(\varOmega )\).
Proof
To prove (4.1), we take \(h\in {\mathbb {R}}^N{\setminus }\{0\}\) and \(\varepsilon >0\), then there exists \(v_{h,\varepsilon }\in C^\infty _0(\varOmega )\) such that
Thus, for \(h\not =0\) we getFootnote 1
By using (4.2), we then obtain
We now integrate with respect to \(h\in {\mathbb {R}}^N\) and use spherical coordinates. This yields
By making the change of variable \(t/2=\tau \) and exploiting the arbitrariness of \(\varepsilon >0\), we eventually reach the desired estimate. \(\square \)
Corollary 4.2
(Interpolation inequality for \({\mathcal {D}}^{s,p}_0\)) Let \(1<p<\infty \) and \(0<s<1\). Let \(\varOmega \subset {\mathbb {R}}^N\) be an open set. For every \(u\in C^\infty _0(\varOmega )\) we have
Proof
It is sufficient to combine Propositions 4.1 and 3.2. \(\square \)
Remark 4.3
For \(p\searrow 1\), the previous inequality becomes [7, Proposition 4.2]. In this case, the constant in (4.3) is sharp for \(N=1\).
For a general open set \(\varOmega \subset {\mathbb {R}}^N\), the converse of inequality (4.1) does not hold. This means that in general we have
the inclusion being continuous. We use the construction of “Appendix A”, in order to give a counterexample.
Example 4.4
With the notation of “Appendix A”, let us takeFootnote 2
For every \(\varepsilon >0\), we take \(u_n\in C^\infty _0({\widetilde{\varOmega }}_n)\subset C^\infty _0(E)\) such that
Here the set \({\widetilde{\varOmega }}_n\) is defined by
On the other hand, by Corollary 3.3 we have
where we also used (A.4), to infer that \(\lambda ^1_p(E)>0\). By Lemma A.1, we have that \(\lambda ^s_{p}(\varOmega _n)\) converges to 0 for \(s\,p<1\), so that
Since \(\varepsilon >0\) is arbitrary, we obtain
4.2 Convex sets
We now prove the converse of (4.1), under suitable assumptions on \(\varOmega \). We start with the case of a convex set. The case \(\varOmega ={\mathbb {R}}^N\) is simpler and instructive; thus, we give a separate statement. The proof can be found, for example, in [28, Lemma 35.2]. We reproduce it, for the reader’s convenience. We also single out an explicit determination of the constant.
Proposition 4.5
(Comparison of norms II: \({\mathbb {R}}^N\)) Let \(1<p<\infty \) and \(0<s<1\). For every \(u\in C^\infty _0({\mathbb {R}}^N)\) we have
In particular, we have that \({\mathcal {D}}^{s,p}_0({\mathbb {R}}^N)={\mathcal {X}}^{s,p}_0({\mathbb {R}}^N)\).
Proof
Let \(u\in C^\infty _0({\mathbb {R}}^N)\), we set
and observe that by construction
We also define
thus, by Jensen’s inequality we have
We now take the compactly supported Lipschitz function
where \((\,\cdot \,)_+\) stands for the positive part. Observe that \(\psi \) has unit \(L^1\) norm, by construction. We then define
By observing that \(\psi _t*u\in C^\infty _0({\mathbb {R}}^N)\), from the definition of the K-functional, we get
We estimate the two norms in the right-hand side separately: for the first one, by Minkowski inequality and Fubini Theorem we get
For the norm of the gradient, we first observe that
thus, we can write
Consequently, by Minkowski inequality we get
In conclusion, we obtained for every \(t>0\)
If we integrate on (0, T), the previous estimate gives
If we now use Lemma B.1 with \(\alpha =p+s\,p\) for the function
we get
where we used (4.4) in the second inequality. By letting T going to \(+\infty \), we get the desired estimate. \(\square \)
Remark 4.6
(Extensions by zero in \({\mathcal {X}}^{s,p}_0(\varOmega )\)...reprise) We take the set \(E\subset {\mathbb {R}}^N\) and the sequence \(\{u_n\}_{n\in {\mathbb {N}}}\subset C^\infty _0(E)\) as in Example 4.4. We have seen that
By observing that
and using Proposition 4.5, we obtain
as well, still for \(s\,p<1\). This shows that the “extension by zero” operator
is not an isometry and, even worse, the two norms
are not equivalent on \(C^\infty _0(E)\). This is in contrast with the case of \(L^p(E)\), \(\mathcal {D}^{1,p}_0(E)\) and \(\mathcal {D}^{s,p}_0(E)\).
We denote by
the inradius of an open set \(\varOmega \subset {\mathbb {R}}^N\). This is the radius of the largest open ball inscribed in \(\varOmega \). We introduce the eccentricity of an open bounded set \(\varOmega \subset {\mathbb {R}}^N\), defined by
Observe that this is a scaling invariant quantity. By generalizing the construction used in [9, Lemma A.6] for a ball, we have the following.
Theorem 4.7
(Comparison of norms II: bounded convex sets) Let \(1<p<\infty \) and \(0<s<1\). If \(\varOmega \subset {\mathbb {R}}^N\) is an open bounded convex set, then for every \(u\in C^\infty _0(\varOmega )\) we have
for a constant \(C=C(N,p,{\mathcal {E}}(\varOmega ))>0\), which blows up as \({\mathcal {E}}(\varOmega )\rightarrow +\infty \). In particular, we have \({\mathcal {X}}^{s,p}_0(\varOmega )={\mathcal {D}}^{s,p}_0(\varOmega )\).
Proof
The proof runs similarly to that of Proposition 4.5 for \({\mathbb {R}}^N\), but now we have to pay attention to boundary issues. Indeed, the function \(\psi _t*u\) is not supported in \(\varOmega \), unless t is sufficiently small, depending on u itself. In order to avoid this, we need to perform a controlled scaling of the function. By keeping the same notation as in the proof of Proposition 4.5, we need the following modification: we take a point \(x_0\in \varOmega \) such that
Without the loss of generality, we can assume that \(x_0=0\). Then we define the rescaled function
We observe that
and by Lemma C.1, we have
This implies that
We can now estimate the K-functional by using the choice \(v=\psi _t*u_t\), that is
Let us set
then we have that for every \(x\in \varOmega \),
By using this and Jensen’s inequality, we obtain
Thus, by using a change of variable and Fubini theorem, we get
where we used that
We now observe that
thus, in particular,
i.e., for every \(x\in \varOmega \) and \(z\in {\widetilde{\varOmega }}\),
This implies that for \(x\in \varOmega \) and \(z\in {\widetilde{\varOmega }}\) we get
Thus, we obtain
Observe that by construction
We now need to show that
We first observe that
and by the divergence theorem
Thus, we obtain as well
and by Hölder’s inequality
for every \(0<t<R_\varOmega /2\). This yields
As above, we now observe that
thus, in particular, for \(0<t<R_\varOmega /2\) we have
This implies that for \(z,w\in {\mathbb {R}}^N\) we have
By inserting this estimate in (4.9), we now get (4.8). This and (4.7) then give
for a constant \(C=C(N,p,{\mathcal {E}}(\varOmega ))\).
We are left with estimating the integral of the K-functional on \((R_\varOmega /2,+\infty )\): for this, we can use the trivial decomposition
which gives
where we used the Poincaré inequality for \({\mathcal {D}}^{s,p}_0(\varOmega )\). By recalling that for a convex set with finite inradius, we have (see [8, Corollary 5.1])
for a constant \({\mathcal {C}}={\mathcal {C}}(N,p)>0\), we finally obtain
By using this in conjunction with (4.10), we finally conclude the proof. \(\square \)
For general unbounded convex sets, the previous proof does not work anymore. However, for convex cones the result still holds. We say that a convex set \(\varOmega \subset {\mathbb {R}}^N\) is a convex cone centered at\(x_0\in {\mathbb {R}}^N\) if for every \(x\in \varOmega \) and \(\tau >0\), we have
Then we have the following
Corollary 4.8
(Comparison of norms II: convex cones) Let \(1<p<\infty \) and \(0<s<1\). If \(\varOmega \subset {\mathbb {R}}^N\) is an open convex cone centered at \(x_0\in {\mathbb {R}}^N\), then for every \(u\in C^\infty _0(\varOmega )\) we have
for a constant \(C=C(N,p,{\mathcal {E}}(\varOmega \cap B_1(x_0)))>0\). In particular, we have \({\mathcal {X}}^{s,p}_0(\varOmega )={\mathcal {D}}^{s,p}_0(\varOmega )\).
Proof
We assume for simplicity that \(x_0=0\) and take \(u\in C^\infty _0(\varOmega )\). Since u has compact support, we have that \(u\in C^\infty _0(\varOmega \cap B_R(0))\), for R large enough. From Theorem 4.7, we know that
We recall that the constant C depends on the eccentricity of \(\varOmega \cap B_R(0)\). However, since \(\varOmega \) is a cone, we easily get
i.e., the constant C is independent of R. Finally, by observing that
we get the desired conclusion. \(\square \)
Remark 4.9
(Rotationally symmetric cones) Observe that if \(\varOmega \) is the rotationally symmetric convex cone
we have
by elementary geometric considerations.
In particular, when \(\varOmega \) is a half-space (i.e., when \(\beta =0\)), then we have \({\mathcal {E}}(\varOmega \cap B_1(0))=2\).
4.3 Lipschitz sets and beyond
In this section, we show that the norms of \({\mathcal {X}}^{s,p}_0\) and \({\mathcal {D}}^{s,p}_0\) are equivalent on open bounded Lipschitz sets. We also make some comments on more general sets, see Remark 4.11.
By generalizing the idea of [22, Theorem 11.6] (see also [6, Theorem 2.1]) for \(p=2\) and smooth sets, we can rely on the powerful extension theorem for Sobolev functions proved by Stein and obtain the following
Theorem 4.10
(Comparison of norms II: Lipschitz sets) Let \(1<p<\infty \) and \(0<s<1\). Let \(\varOmega \subset {\mathbb {R}}^N\) be an open bounded set, with Lipschitz boundary. Then for every \(u\in C^\infty _0(\varOmega )\) we have
for a constant \(C_1>0\) depending on \(N,p, {\mathrm {diam}}(\varOmega )\) and the Lipschitz constant of \(\partial \varOmega \). In particular, we have \({\mathcal {X}}^{s,p}_0(\varOmega )={\mathcal {D}}^{s,p}_0(\varOmega )\) in this case as well.
Proof
We take an open ball \(B\subset {\mathbb {R}}^N\) with radius \({\mathrm {diam}}(\varOmega )\) and such that \(\varOmega \Subset B\). We then take a linear and continuous extension operator
such that
where \(\mathfrak {e}_\varOmega >0\) depends on \(N,p,{\mathrm {diam}}(\varOmega )\) and the Lipschitz constant of \(\partial \varOmega \). We observe that such an operator exists, thanks to the fact that \(\varOmega \) has a Lipschitz boundary, see [27, Theorem 5, p. 181]. We also observe that the first estimate in (4.11) is not explicitly stated by Stein, but it can be extrapolated by having a closer look at the proof, see [27, p. 192].
For every \(v\in C^\infty _0(B)\), we define the operator
and observe that
Since \(\varOmega \) has continuous boundary, this implies that \({\mathcal {R}}(v)\in {\mathcal {D}}^{1,p}_0(\varOmega )\), see Remark 2.1. We now fix \(u\in C^\infty _0(B)\), for every \(v\in C^\infty _0(B)\) and every \(t,\varepsilon >0\), we take \(\varphi _{\varepsilon ,t}\in C^\infty _0(\varOmega )\) such that
This is possible, thanks to the definition of \({\mathcal {D}}^{1,p}_0(\varOmega )\). Then for \(t>0\) we can estimate the relevant K-functional as follows
By applying (4.11) and using that
we then get
We now use that
thanks to Poincaré inequality. By spending this information in the previous estimate and using the arbitrariness of \(\varepsilon \), we get
We set for simplicity
then by taking the infimum over \(v\in C^\infty _0(B)\)
As usual, we integrate in t, so to get
We now observe that if \(u\in C^\infty _0(\varOmega )\), then we have \({\mathcal {R}}(u)=u\). Thus, from (4.12) and Theorem 4.7 for the convex set B, we get
where C only depends on N and p. This concludes the proof. \(\square \)
Remark 4.11
(More general sets) It is not difficult to see that the previous proof works (and thus \({\mathcal {X}}^{s,p}_0(\varOmega )\) and \({\mathcal {D}}^{s,p}_0(\varOmega )\) are equivalent), whenever the set \(\varOmega \) is such that there exists a linear and continuous extension operator
such that (4.11) holds. Observe that there is a vicious subtlety here: the first condition in (4.11) is vital and, in general, it may fail to hold for an extension operator. For example, there is a beautiful extension result by Jones [21, Theorem 1], which is valid for very irregular domains (possibly having a fractal boundary): however, the construction given by Jones does not assure that the first estimate in (4.11) holds true, see the statement of [21, Lemma 3.2].
In order to complement the discussion of Remarks 3.4 and 4.6 on “extensions by zero” in \({\mathcal {X}}^{s,p}_0\), we explicitly state the following consequence of Proposition 4.1 and Theorem 4.10.
Corollary 4.12
Let \(1<p<\infty \) and \(0<s<1\). Let \(\varOmega \subset {\mathbb {R}}^N\) be an open bounded set, with Lipschitz boundary. Then for every \(u\in C^\infty _0(\varOmega )\), we have
where \(C_1>0\) is the same constant as in Theorem 4.10.
5 Capacities
Let \(1<p<N\), we recall that for every compact set \(F\subset {\mathbb {R}}^N\), its p-capacity is defined by
see [17, Chapter 4, Section 7].
Similarly, given \(1<p<\infty \) and \(0<s<1\) such thatFootnote 3\(s\,p<N\), we define the (s, p)-capacity ofF through
and the interpolation (s, p)-capacity ofF by
As a straightforward consequence of Propositions 4.1 and 4.5, we have the following
Corollary 5.1
(Comparison of capacities) Let \(1<p<\infty \) and \(0<s<1\) be such that \(s\,p<N\). Let \(F\subset {\mathbb {R}}^N\) be a compact set, then we have
for a constant \(C=C(N,p)>1\). In particular, it holds
Proposition 5.2
Let \(1<p<\infty \) and \(0<s<1\) be such that \(s\,p<N\). For every \(E,F\subset {\mathbb {R}}^N\) compact sets, we have
Proof
We fix \(n\in {\mathbb {N}}{\setminus }\{0\}\) and choose two nonnegative functions \(\varphi _n,\psi _n\in C^\infty _0({\mathbb {R}}^N)\) such that
and
We then set
where \(\{\varrho _\varepsilon \}_{\varepsilon >0}\) is a family of standard Friedrichs mollifiers. We observe that for every \(n\in {\mathbb {N}}{\setminus }\{0\}\), it holds that \(U_{n,\varepsilon }\in C^\infty _0({\mathbb {R}}^N)\). Moreover, by construction we have
By observing that Jensen’s inequality implies
we thus get
By using the sub-modularity of the Sobolev–Slobodeckiĭ seminorm (see [19, Theorem 3.2 & Remark 3.3]), we obtain
Finally, thanks to the choice of \(\varphi _n\) and \(\psi _n\), we get the desired conclusion by the arbitrariness of n. \(\square \)
In the next result, we denote by \({\mathcal {H}}^\tau \) the \(\tau \)-dimensional Hausdorff measure.
Proposition 5.3
Let \(1<p<\infty \) and \(0<s<1\) be such that \(s\,p<N\). Let \(\varOmega \subset {\mathbb {R}}^N\) be an open set. We take a compact set \(E\Subset \varOmega \) such that
Then we have
and
Proof
To prove (5.1), we can easily adapt the proof of [17, Theorem 4, p. 156], dealing with the local case.
In order to prove (5.2), we first assume \(\varOmega \) to be bounded. Let \(\varepsilon >0\), we take \(u_\varepsilon \in C^\infty _0(\varOmega )\) such that
We further observe that the boundedness of \(\varOmega \) implies that
and that any solution \(u\in {\mathcal {D}}^{s,p}_0(\varOmega )\) has norm \(L^\infty (\varOmega )\) bounded by a constant \(M=M(N,s,p,\varOmega )\), see [7, Theorem 3.3]. Thus, without the loss of generality, we can also assume that
Since E has null (s, p)-capacity, there exists \(\varphi _\varepsilon \in C^\infty _0(\varOmega )\) such that
We set \(\psi _\varepsilon =\varphi _\varepsilon /\Vert \varphi _\varepsilon \Vert _{L^\infty ({\mathbb {R}}^N)}\) and observe that \(\Vert \varphi _\varepsilon \Vert _{L^\infty ({\mathbb {R}}^N)}\ge 1\). The function \(u_\varepsilon \,(1-\psi _\varepsilon )\) is admissible for the variational problem defining \(\lambda ^s_p(\varOmega {\setminus } E)\); then by using the triangle inequality, we have
From the first part of the proof, we know that E has N-dimensional Lebesgue measure 0; thus, the \(L^p\) norm over \(\varOmega {\setminus } E\) is the same as that over \(\varOmega \). If we now take the limit as \(\varepsilon \) goes to 0 and use the properties of \(u_\varepsilon \), together withFootnote 4
and
we get
The reverse inequality simply follows from the fact that \(C^\infty _0(\varOmega {\setminus } E)\subset C^\infty _0(\varOmega )\); thus, we get the conclusion when \(\varOmega \) is bounded.
In order to remove the last assumption, we consider the sets \(\varOmega _R=\varOmega \cap B_R(0)\). For R large enough, this is a nonempty open bounded set and \(E\Subset \varOmega _R\) as well. We thus have
By taking the limitFootnote 5 as R goes to \(+\infty \), we get the desired conclusion in the general case as well. \(\square \)
The previous result giving the link between the Poincaré constant and sets with null capacity does not hold true in the interpolation space \({\mathcal {X}}^{s,p}_0(\varOmega )\). Indeed, we have the following result, which shows that the interpolation Poincaré constant is sensitive to removing sets with null (s, p)-capacity.
Lemma 5.4
Let \(1<p<N\) and \(0<s<1\). Let \(\varOmega \subset {\mathbb {R}}^N\) be an open set and \(E\Subset \varOmega \) a compact set such that
Then we have
Proof
By Corollary 3.3, we know that
It is now sufficient to use that \(\lambda ^1_p(\varOmega {\setminus } E)>\lambda ^1_p(\varOmega )\), as a consequence of the fact that E has positive p-capacity. \(\square \)
Remark 5.5
As an explicit example of the previous situation, we can take \(s\,p<1\) and the \((N-1)\)-dimensional set
Observe that \({\mathrm {cap}}_p(F)>0\) by [17, Theorem 4, p. 156]. On the other hand, we have
Indeed, we set
We then take the usual sequence of Friedrichs mollifiers \(\{\varrho _\varepsilon \}_{\varepsilon >0}\subset C^\infty _0({\mathbb {R}}^N)\) and define
Observe that by construction we have
By definition of (s, p)-capacity and using the interpolation estimate (4.3), we get
We then observe that the last quantity goes to 0 as \(\varepsilon \) goes to 0, thanks to the fact that \(s\,p<1\). By Corollary 5.1, we have
as desired.
6 Double-sided estimates for Poincaré constants
We already observed that for an open set \(\varOmega \subset {\mathbb {R}}^N\) we have
We now want to compare \(\lambda ^1_p\) with the sharp Poincaré constant for the embedding \({\mathcal {D}}^{s,p}_0(\varOmega )\hookrightarrow L^p(\varOmega )\).
Theorem 6.1
Let \(1<p<\infty \) and \(0<s<1\). Let \(\varOmega \subset {\mathbb {R}}^N\) be an open set, then
If in addition:
-
\(\varOmega \subset {\mathbb {R}}^N\) is bounded with Lipschitz boundary, then we also have the reverse inequality
$$\begin{aligned} \frac{1}{p\,C_1}\,\Big (\lambda ^1_{p}(\varOmega )\Big )^s\le s\,(1-s)\,\lambda ^s_{p}(\varOmega ), \end{aligned}$$(6.2)where \(C_1>0\) is the same constant as in Theorem 4.10;
-
\(\varOmega \subset {\mathbb {R}}^N\) is convex, then we also have the reverse inequality
$$\begin{aligned} \frac{1}{C_2}\,\Big (\lambda ^1_{p}(\varOmega )\Big )^s\le s\,(1-s)\,\lambda ^s_{p}(\varOmega ), \end{aligned}$$(6.3)where \(C_2\) is the universal constant given by
$$\begin{aligned} C_2=\frac{\Big (\lambda ^1_p(B_1(0))\Big )^s}{{\mathcal {C}}}, \end{aligned}$$and \({\mathcal {C}}={\mathcal {C}}(N,p)>0\) is the same constant as in the Hardy inequality for \({\mathcal {D}}^{s,p}_0(\varOmega )\) (see [8, Theorem 1.1]).
Proof
The first inequality (6.1) is a direct consequence of the interpolation inequality (4.3). Indeed, by using the definition of \(\lambda ^s_{p}(\varOmega )\), we obtain from this inequality
for every \(u\in C^\infty _0(\varOmega )\). By simplifying the factor \(\Vert u\Vert ^p_{L^p(\varOmega )}\) on both sides and taking the infimum over \(C^\infty _0(\varOmega )\), we get the claimed inequality.
In order to prove (6.2), for every \(\varepsilon >0\) we take \(\varphi \in C^\infty _0(\varOmega )\) such that
then we use Theorem 4.10 to infer
This in turn implies
by arbitrariness of \(\varepsilon >0\). A further application of Corollary 3.3 leads to the desired conclusion.
Finally, if \(\varOmega \subset {\mathbb {R}}^N\) is convex, we can proceed in a different way. We first observe that we can always suppose that the inradius \(R_\varOmega \) is finite; otherwise, both \(\lambda ^1_p(\varOmega )\) and \(\lambda ^s_p(\varOmega )\) vanish, and there is nothing to prove. Then (6.3) comes by joining the simple estimate
which follows from the monotonicity and scaling properties of \(\lambda ^1_p\), and the estimate of [8, Corollary 5.1], i.e.,
The latter is a consequence of the Hardy inequality in convex sets for \({\mathcal {D}}^{s,p}_0\). \(\square \)
Remark 6.2
For \(p=2\), the double-sided estimate of Theorem 6.1 is contained in [12, Theorem 4.5]. The proof in [12] relies on probabilistic techniques, and the result is proved by assuming that \(\varOmega \) verifies a uniform exterior cone condition.
Remark 6.3
Inequality (6.2) cannot hold for a general open set \(\varOmega \subset {\mathbb {R}}^N\), with a constant independent of \(\varOmega \). Indeed, one can construct a sequence \(\{\varOmega _n\}_{n\in {\mathbb {N}}}\subset {\mathbb {R}}^N\) such that
see Lemma A.1.
Notes
In the second inequality, we use the classical fact
$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N} |\varphi (x+h)-\varphi (x)|^{p}\,\mathrm{d}x&=\int _{{\mathbb {R}}^N} \left| \int _0^1\langle \nabla \varphi (x+t\,h),h\rangle \,\mathrm{d}t\right| ^{p}\,\mathrm{d}x\\&\le |h|^p\,\int _{{\mathbb {R}}^N}\int _0^1 |\nabla \varphi (x+t\,h)|^p\,\mathrm{d}t\,\mathrm{d}x\\&= |h|^p\,\int _0^1 \left( \int _{{\mathbb {R}}^N}|\nabla \varphi (x+t\,h)|^p\,\mathrm{d}x\right) \,\mathrm{d}t=|h|^p\,\Vert \nabla \varphi \Vert _{L^p({\mathbb {R}}^N)}. \end{aligned} \end{aligned}$$In dimension \(N=1\), we simply take \(E={\mathbb {R}}{\setminus }{\mathbb {Z}}\).
As usual, the restriction \(s\,p<N\) is due to the scaling properties of the relevant energies. It is not difficult to see that for \(s\,p\ge N\), both infima are identically 0.
Observe that, from the first condition, we get that \(\psi _\varepsilon \) converges to 0 strongly in \(L^p(\varOmega )\), by Sobolev inequality. Since the family \(\{u_\varepsilon \}\) is bounded in \(L^\infty (\varOmega )\), this is enough to infer
$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \int _{\varOmega } |u_\varepsilon |^p\,|1-\psi _\varepsilon |^p\,\mathrm{d}x=1. \end{aligned}$$Such a limit exists by monotonicity.
For \(N=1\), the set F simply coincides with the point \(\{0\}\).
For u, v differentiable functions with \(v\ge 0\) and \(u>0\), we have the pointwise inequality
$$\begin{aligned} |u'|^{p-2}\,u'\,\left( \frac{v^p}{u^{p-1}}\right) '\le |v'|^p. \end{aligned}$$
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Acknowledgements
The first author would like to thank Yavar Kian and Antoine Lemenant for useful discussions on Stein’s and Jones’ extension theorems. Simon Chandler-Wilde is gratefully acknowledged for some explanations on his paper [11]. This work started during a visit of the second author to the University of Ferrara in October 2017.
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Appendices
Appendix A. An example
In this section, we construct a sequence of open bounded sets \(\{\varOmega _n\}_{n\in {\mathbb {N}}}\subset {\mathbb {R}}^N\) with rough boundaries and fixed diameter, such that we have
The sets \(\varOmega _n\) are obtained by removing from an N-dimensional cube an increasing array of regular \((N-1)\)-dimensional cracks (Fig. 1).
For \(N\ge 1\), we setFootnote 6
For every \(n\in {\mathbb {N}}\), we also define
Finally, we consider the sets
and
Then (A.1) is a consequence of the next result.
The set \(\varOmega _n\) in dimension \(N=2\), for \(n=3\)
Lemma A.1
With the notation above, for \(1<p<\infty \) and \(s<1/p\) we have
and
In particular, the new sequence of rescaled sets \(\{\varOmega _n\}_{n\in {\mathbb {N}}}\subset {\mathbb {R}}^N\) defined by
is such that
Proof
We divide the proof in two parts, for ease of readability. Of course, it is enough to prove (A.2) and (A.3). Indeed, the last statement is a straightforward consequence of these facts and of the scaling properties of the diameter and of the Poincaré constants. \(\square \)
Proof of (A.2)
For \(1<p<\infty \) we define
We first observe that F is a compact set with positive \((N-1)\)-dimensional Hausdorff measure; thus, by [23, Theorem 10.1.2] we can infer the existence of a constant \(C=C(N,p,F)>0\) such that
This shows that \(\mu _p(Q;F)>0\).
For every \(\varepsilon >0\), we consider \(u_{\varepsilon }\in C_0^\infty (E){\setminus }\{0\}\) such that
We now observe that for every \(z\in {\mathbb {Z}}^N\), there holds
thanks to the fact that \(u_\varepsilon \) vanishes on (the relevant translated copy of) F and to the fact that \(\mu _p(Q,F)=\mu _p(Q+z,F+z)\). By using this information, we get
By recalling the choice of \(u_\varepsilon \), we then get
Thanks to the arbitrariness of \(\varepsilon >0\) and to the fact that \({\widetilde{\varOmega }}_n\subset E\), this finally gives
as desired. \(\square \)
Proof of (A.3)
We recall that
and that each \((N-1)\)-dimensional set \(F+z\) has null (s, p)-capacity, thanks to Remark 5.5. By using Proposition 5.2, we also obtain
Then by Proposition 5.3, we get
This is turn gives the desired conclusion (A.3). \(\square \)
Appendix B. One-dimensional Hardy inequality
In the proof of Proposition 4.5, we used the following general form of the one-dimensional Hardy inequality. (The classical case corresponds to \(\alpha =p-1\) below.) This can be found, for example, in [23, p. 39]. For the sake of completeness, we give a sketch of a proof based on Picone’s inequality.Footnote 7
Lemma B.1
Let \(1<p<\infty \) and \(\alpha >0\). For every \(f\in C^\infty _0((0,T])\) we have
Proof
We take \(0<\beta <\alpha /(p-1)\) and consider the function \(\varphi (t)=t^\beta \). Observe that this solves
Thus, for every \(\psi \in C_0^\infty ((0,T])\) we have the weak formulation
We take \(\varepsilon >0\) and \(f\in C^\infty _0((0,T])\) nonnegative, we insert the test function
in the previous integral identity. By using Picone’s inequality, we then obtain
If we take the limit as \(\varepsilon \) goes to 0, by Fatou’s lemma we get
The previous inequality holds true for every \(0<\beta <\alpha /(p-1)\) and \(\beta ^{p-1}\,(\alpha -\beta \,(p-1))\) is maximal for \(\beta =\alpha /p\). This concludes the proof. \(\square \)
Appendix C. A geometric lemma
When comparing the norms of \({\mathcal {X}}^{s,p}_0(\varOmega )\) and \({\mathcal {D}}^{s,p}_0(\varOmega )\) for a convex set, we used the following geometric result. We recall that
is the inradius of\(\varOmega \), i.e., the radius of the largest ball inscribed in \(\varOmega \).
Lemma C.1
Let \(\varOmega \subset {\mathbb {R}}^N\) be an open convex set such that \(R_\varOmega <+\infty \). Let \(x_0\in \varOmega \) be a point such that
Then for every \(0<t<1\) we have
Proof
Without the loss of generality, we can assume that \(0\in \varOmega \) and that \(x_0=0\). Clearly, it is sufficient to prove that
Every point of \(\partial (t\,\varOmega )\) is of the form \(t\,z\), with \(z\in \partial \varOmega \). We now take the cone \(C_z\), obtained as the convex envelope of \(B_{R_\varOmega }(0)\) and the point z. By convexity of \(\varOmega \), we have of course \(C_z\subset \varOmega \). We thus obtain
We now distinguish two cases:
-
(i)
\(|z|=R_\varOmega \);
-
(ii)
\(|z|>R_\varOmega \).
When alternative (i) occurs, then \(C_z=B_{R_\varOmega }(0)\) and thus
By using this in (C.1), we get the desired estimate.
If on the contrary we are in case (ii), then by elementary geometric considerations we have
see Fig. 2. This gives again the desired conclusion. \(\square \)
The case (ii) in the proof of Lemma C.1. Colored in red, the distance of \(t\,z\) from \(\partial C_z\) (color figure online)
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Brasco, L., Salort, A. A note on homogeneous Sobolev spaces of fractional order. Annali di Matematica 198, 1295–1330 (2019). https://doi.org/10.1007/s10231-018-0817-x
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DOI: https://doi.org/10.1007/s10231-018-0817-x