Equivalence between validity of the $p$-Poincaré inequality and finiteness of the strict $p$-capacitary inradius.

Let $\Omega$ be an open set in $\mathbb{R}^{n}$, and $\|\,.\,\|_{p,\Omega}$ the $L^{p}$-norm on $\Omega$ for $1<p<\infty$. The classical $p$-Poincaré inequality is said to hold on $\Omega$ if there exists a constant $C>0$ such that

where $\mathcal{C}_{c}^{\infty}(\Omega)$ denotes the space of smooth functions with compact support in $\Omega$. That is, (1.1) holds if and only if the infimum of the so-called nonlinear Rayleigh quotient

is positive. If $\lambda_{1,p}(\Omega)$ is positive and attained at some non-zero function in $W^{1,p}_{0}(\Omega)$, i.e., the closure of $\mathcal{C}_{c}^{\infty}(\Omega)$ with respect to the $L^{p}(\Omega)$-Sobolev 1-norm, then it is the smallest generalized eigenvalue for the $p$-Laplacian, with Dirichlet boundary condition, on $\Omega$ in the distributional sense.

The validity of (1.1), or equivalently the positivity of $\lambda_{1,p}(.)$, plays a role in establishing existence and uniqueness of solutions to certain quasilinear elliptic equations, see [20, Theorem 1.2] as well as in determining the asymptotic behavior of solutions to some nonlinear parabolic equations, see [18, Sections 3&4]. Furthermore, in the case of $n=p$, it was recently shown in [5, Theorems 1&2] that a local version of the Moser–Trudinger inequality implies the global Moser–Trudinger inequality if and only (1.1) holds.

In this note, we employ the concept of strict $p$-capacitary inradius of $\Omega$, originally introduced in the case of $p=2$ in [7] for $n=2$ and in [6] for $n\geq 3$, and derive the equivalence between its finiteness and the positivity of $\lambda_{1,p}(\Omega)$. In the definition of the strict $p$-capacitary inradius, we use the notion of Sobolev $p$-capacity, $C_{p}(K)$, which, for a compact set $K\subset\mathbb{R}^{n}$, is defined as

where $\mathcal{S}$ denotes the space of Schwartz functions. The strict $p$-capacitary inradius of $\Omega$ is defined as

where $\mathbb{B}_{r}(x)$ denotes the open ball in $\mathbb{R}^{n}$ of radius $r>0$ with center $x\in\mathbb{R}^{n}$. Roughly speaking, finiteness of $\rho_{p}(\Omega)$ means that the complement of $\Omega$ is somewhat evenly und uniformly distributed in $\mathbb{R}^{n}$. In fact, if $\rho_{p}(\Omega)$ is finite and $R>\rho_{p}(\Omega)$, then there exists a $\delta>0$ such that within $R$-units of any point in $\mathbb{R}^{n}$, we may find a set in the complement of $\Omega$ whose Sobolov $p$-capacity is at least $\delta$.

Finiteness of $\rho_{p}(\Omega)$ is a necessary condition for (1.1) to hold on an open set $\Omega$. Indeed, our first result yields a sharp upper bound for $\rho_{p}(\Omega)$ in terms of $\lambda_{1,p}(\Omega)$.

The heart of the matter of the proof of Theorem 1.3 is a continuity result for $\lambda_{1,p}$. That is, suppose $\{K_{j}\}_{j\in\mathbb{N}}$ is a sequence of compact sets contained in the closure of some bounded domain $D$ with smooth boundary. If $C_{p}(K_{j})$ tends to zero as $j\to\infty$ and, for each $j\in\mathbb{N}$, $D\setminus K_{j}$ is an open set with smooth boundary, then $\lambda_{1,p}(D\setminus K_{j})$ tends to $\lambda_{1,p}(D)$. This was orginally proved in [7] for the case $n=2$, $p=2$, $D=\mathbb{B}_{1}(0)$, and the logarithmic capacity in place of $C_{p}$. Regularity results for the Dirichlet problem and for first eigenfunctions of the $p$-Laplacian, $\Delta_{p}$, on open, bounded sets with smooth boundary as well as the theory of $p$-harmonic functions allow this continuity result to be extended to the case of the Sobolev $p$-capacity with $1<p<\infty$, see Lemma 3.1.

To state the sufficiency of the finiteness of $\rho_{p}(\Omega)$, we define the scalar

for any $R>0$. Note that $\delta_{R}(\Omega)$ is positive whenever $\rho_{p}(\Omega)$ is finite and $R>\rho_{p}(\Omega)$.

Part (i) of Theorem 1.6 is a direct consequence of a Poincaré-type inequality for any function in $W^{1,p}(\Omega)$ which has a representative that vanishes on a set of positive $p$-Sobolev capacity in $\Omega$, where $\Omega$ is a bounded extension domain. This inequality originates in the work of Meyers [17], and was completed by Adams, see Theorem 8.3.3 and the notes in Section 8.3 on pg. 231 in [1]. For the proof of Theorem 1.6, one simply writes $\mathbb{R}^{n}$ as a union of closed cubes with mutually disjoint interiors and of side length larger than the strict $p$-capacitary inradius of the open set under consideration. Then, one applies this Poincaré-type inequality to each cube. This kind of proof is contained in the works of Maz’ya– Shubin [14] and Souplet [18]. Souplet also splits $\mathbb{R}^{n}$ into cubes and then uses a Poincaré-type inequality for functions which vanish on a set of positive Lebesgue measure. Maz’ya–Shubin derive a Poincaré-type inequality in [14, Lemma 3.1], similar to the one used in this note, but for balls which forces them take the multiplicity of coverings by balls into account in order to obtain a global estimate.

Theorems 1.3 and 1.6 may be summarized in a qualitative manner as follows.

It is well-known that the finiteness of the inradius, $\mathfrak{R}(\Omega)$, of an open set $\Omega$, i.e., the supremum of the radii of all balls contained in $\Omega$, is a necessary condition for (1.1). This can be seen by a simple scaling argument, see, e.g., Souplet’s proof of [18, Prop. 2.1 (i)]. Souplet additionally shows in [18, Prop. 2.1] with $p\in[1,\infty)$, that this condition is also sufficient as long as $\Omega$ is a domain which satisfies a uniform exterior cone condition; in the case of $p=2$ this appears to go back to work of Agmon [2]. In [18], Souplet introduces a measure-theoretic inradius, which yields a sufficient condition for the validity of (1.1) without any regularity assumptions on the boundary of the domain $\Omega$. Souplet’s formulation of inradius actually inspired the notion of the capacitary inradius in (1.2), originally introduced in [7]. A sufficient condition, similar to Souplet’s, was previously obtain by Lieb in [10, Corollary 2]. We note that the assumption of finiteness of either of these conditions is a stronger assumption than the one of finiteness of the strict $p$-capacitary inradius defined in (1.2). The reason for that is that these measure-theoretic inradii do not take into account sets in the complement which are of Lebesgue measure zero but of positive Sobolev $p$-capacity. However, for any pair $(p^{\prime},p)$ with $1<p^{\prime}<p\leq n$, there exists a set $E\subset\mathbb{R}^{n}$ of Lebesgue measure zero, such that $C_{p^{\prime}}(E)=0$ while $C_{p}(E)>0$, see [1, Theorem 5.5.1].

A complete description in the flavor of Corollary 1.9 was first given by Maz’ya–Shubin in [14] in the case of $p=2$ and $n\geq 3$. The authors of [14] use different notions of capacity and of capacitary inradius than presented in this note; see Lemma 2.14 on how their capacitary inradius relates to the one defined in (1.2). For $p=2$ and $n\geq 3$, estimate (1.4) is an improvement over the upper bounded for $\lambda_{1,2}(\Omega)$ provided in [14] while (1.8) and the lower bound given in [14] are similar. We also point to the work of Vitolo [20] in which he shows that if $p>n$ and $\Omega$ is a domain with finite inradius, $\mathfrak{R}(\Omega)$, then $\lambda_{1,p}(\Omega)>0$. Note that if $p>n$, then singletons have positive $p$-Sobolev capacity, so that $\mathfrak{R}(\Omega)=\rho_{p}(\Omega)$, i.e., Theorem 1.6 rediscovers Vitolo’s result.

Together with Lebl and Ramachandran, we considered the problem of describing the validity of the Poincaré inequality in the case of $n=2$, $p=2$ in potential-theoretic terms in [7]. Originally, we intended to investigate which potential-theoretic conditions yield the $L^{2}$-closed range property of (the weak maximal extension of) the Cauchy–Riemann operator which constitutes an open problem in several complex variables. This closed range property turns out to be equivalent to (1.1) on any open set $\Omega\subset\mathbb{R}^{2}$. Moreover, we showed that Corollary 1.9 holds for $p=2$ and that (1.1) is equivalent to the existence of a smooth, bounded function on $\Omega$ such that its Laplacian has a positive lower bound on $\Omega$. These results were later shown to hold for $p=2$, $n\geq 3$ with the Newtonian capacity in place of $C_{p}$ in [6] by the author of this note. We show in Lemma 2.11 that the strict $p$-capacitary inradius defined in (1.2) does not depend on the choice of $p$-capacity as long as the sets of zero $p$-capacity are the same as the sets of for which $C_{p}$ is zero as well. In particular, the strict capacitary inradii defined in [7, 6] are the same as the one defined in (1.2) for $p=2$, see the paragraph subsequent to the proof of Lemma 2.11.

This note is structured as follows. The notions of capacity, strict $p$-capacitary inradius, and $\lambda_{1,p}$ and their basic properties are detailed in Section 2. The proofs of Theorem 1.3 and Theorem 1.6 are given in Section 3 and 4, respectively.

The following lemma is crucial for the proof of Theorem 1.3.