Inequalities between torsional rigidity and principal eigenvalue of the p-Laplacian

Abstract

We consider the torsional rigidity and the principal eigenvalue related to the p-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. The limit cases $p=1$ and $p=\infty$ are also analyzed, which amount to consider the Cheeger constant of a domain and functionals involving the distance function from the boundary.

1 Introduction

In this paper we consider the problem of minimizing or maximizing the quantity

$$\begin{aligned} \lambda _p^\alpha (\Omega )T_p^\beta (\Omega ) \end{aligned}$$

on the class of open sets $\Omega \subset {\mathbb {R}}^d$ having a prescribed Lebesgue measure, where $\alpha ,\beta$ are two real parameters, and $\lambda _p(\Omega )$, $T_p(\Omega )$ are respectively the principal eigenvalue and the torsional rigidity, which are defined below, relative to the p-Laplace operator

$$\begin{aligned} \Delta _p u:={{\,\mathrm{div}\,}}\big (|\nabla u|^{p-2}\nabla u\big ). \end{aligned}$$

In all the paper, we use the following notation:

• $p'$ is the conjugate exponent of p given by $p':=p/(p-1)$;

• $\Omega \subset {\mathbb {R}}^d$ is an open set with finite Lebesgue measure $|\Omega |$;

• $d_\Omega$ is the distance function from $\partial \Omega$

  $$\begin{aligned} d_\Omega (x):=\inf \big \{|x-y|\ :\ y\in \partial \Omega \big \}; \end{aligned}$$

• $\rho (\Omega )$ is the inradius of $\Omega$

  $$\begin{aligned} \rho (\Omega ):=\Vert d_\Omega \Vert _{L^\infty (\Omega )}, \end{aligned}$$

  corresponding to the maximal radius of a ball contained in $\Omega$;

• ${{\,\mathrm{diam}\,}}(\Omega )$ is the diameter of $\Omega$

  $$\begin{aligned} {{\,\mathrm{diam}\,}}(\Omega ):=\sup \big \{|x-y|\ :\ x,y\in \Omega \big \}; \end{aligned}$$

• $P(\Omega )$ is the distributional perimeter of $\Omega$ in the De Giorgi sense, defined by

  $$\begin{aligned} P(\Omega ):=\sup \left\{ \int _\Omega {{\,\mathrm{div}\,}}\phi \,dx\ :\ \phi \in C^1_c({\mathbb {R}}^d;{\mathbb {R}}^d),\ \Vert \phi \Vert _{L^\infty ({\mathbb {R}}^d)}\le 1\right\} ; \end{aligned}$$

• $h(\Omega )$ is the Cheeger constant of $\Omega$, that we define in Sect. 5;

• $B_r$ is the open ball in ${\mathbb {R}}^d$ centered at the origin with radius r and $\omega _d:=|B_1|$;

• ${\mathcal {H}}^{d-1}$ is the $d-1$ dimensional Hausdorff measure.

Given $1<p<\infty$, $T_p(\Omega )$ denotes the p-torsional rigidity of $\Omega$, defined by

$$\begin{aligned} T_p(\Omega )=\max \bigg \{\Big [\int _\Omega |u|\,dx\Big ]^{p}\Big [\int _\Omega |\nabla u|^p\,dx\Big ]^{-1}\ :\ u\in W^{1,p}_0(\Omega ),\ u\ne 0\bigg \}, \end{aligned}$$

(1.1)

where $W^{1,p}_0(\Omega )$ stands for the usual Sobolev space obtained as the completion of the space $C^{\infty }_c(\Omega )$ with respect to the norm $\Vert \nabla u\Vert _{L^p(\Omega )}$. Equivalently, if $w_p$ is the unique weak solution of the nonlinear PDE

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p w=1&{} \hbox {in}\quad \Omega ,\\ w\in W^{1,p}_0(\Omega ), \end{array}\right. } \end{aligned}$$

(1.2)

we can define $T_p(\Omega )$ as (see [7, Proposition 2.2]):

$$\begin{aligned} T_p(\Omega )=\bigg (\int _\Omega w_p dx\bigg )^{p-1}. \end{aligned}$$

(1.3)

Note that $w_p$ is a nonnegative function and (1.2) is the Euler–Lagrange equation of the variational problem

$$\begin{aligned} \min \Big \{J_p(u)\ :\ u\in W^{1,p}_0(\Omega )\Big \}, \end{aligned}$$

where

$$\begin{aligned} J_p(u):=\frac{1}{p}\int _\Omega |\nabla u|^p\,dx-\int _\Omega u\,dx. \end{aligned}$$

(1.4)

Multiplicating by $w_p$ in (1.2) and integrating by parts gives

$$\begin{aligned} \int _\Omega w_p\,dx=\int _\Omega |\nabla w_p|^p\,dx=-p'J_p(w_p). \end{aligned}$$

When $\Omega =B_1$, the solution $w_p$ to the boundary problem (1.2) is explicit and given by

$$\begin{aligned} w_{p}(x)=\frac{1-|x|^{p'}}{p'd^{1/(p-1)}} \end{aligned}$$

(1.5)

which leads to

$$\begin{aligned} T_p(B_1)=\frac{1}{d}\Big (\frac{\omega _d}{p'+d}\Big )^{p-1}. \end{aligned}$$

The p-principal eigenvalue $\lambda _p(\Omega )$ is defined through the Rayleigh quotient

$$\begin{aligned} \lambda _p(\Omega )=\min \left\{ \left[ \int _\Omega |\nabla u|^p\,dx\right] \left[ \int _\Omega |u|^{p}\,dx\right] ^{-1}\ :\ u\in W^{1,p}_0(\Omega ),\ u\ne 0\right\} . \end{aligned}$$

(1.6)

Equivalently, $\lambda _p(\Omega )$ denotes the least value $\lambda$ such that the nonlinear PDE

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p u=\lambda |u|^{p-2}u&{}\hbox {in}\quad \Omega ,\\ u\in W^{1,p}_0(\Omega ), \end{array}\right. } \end{aligned}$$

has a nonzero solution; we recall that in dimension 1 we have (see for instance [22])

$$\begin{aligned} \lambda _p(-1,1)=\left( \frac{\pi _p}{2}\right) ^p\quad \text {where}\quad \pi _p=2\pi \frac{(p-1)^{1/p}}{p\sin (\pi /p)}, \end{aligned}$$

(1.7)

while in higher dimension the following estimate holds true, see [20, Theorem 3.1]:

$$\begin{aligned} \lambda _p(B_1)\le \frac{(p+1)(p+2)\cdots (p+d)}{d!}. \end{aligned}$$

(1.8)

It is easy to see that the two quantities above scale as

$$\begin{aligned} \lambda _p(t\Omega )=t^{-p}\lambda _p(\Omega ),\quad T_p(t\Omega )=t^{p+d(p-1)}T_p(\Omega ). \end{aligned}$$

(1.9)

By using a symmetrization argument and the so-called Pólya–Szegö principle (see [18]) it is possible to prove that balls maximize $T_p$ (respectively minimize $\lambda _p$) among all sets of prescribed Lebesgue measure, which can be written in a scaling free form as

$$\begin{aligned} \lambda _p(B)|B|^{p/d}\le \lambda _p(\Omega )|\Omega |^{p/d},\quad T_p(\Omega )|\Omega |^{1-p-p/d}\le T_p(B)|B|^{1-p-p/d}, \end{aligned}$$

(1.10)

where B is any ball in ${\mathbb {R}}^d$. The inequalities (1.10) are known respectively as Faber–Krahn inequality and Saint-Venant inequality.

Moreover, we have:

$$\begin{aligned}&\inf \big \{T_p(\Omega )\ :\ \Omega \text { open in }{\mathbb {R}}^d,\ |\Omega |=1\big \}=0, \end{aligned}$$

(1.11)

$$\begin{aligned}&\sup \big \{\lambda _p(\Omega )\ :\Omega \text { open in }{\mathbb {R}}^d,\ |\Omega |=1\big \}=+\infty . \end{aligned}$$

(1.12)

To prove (1.11) and (1.12) it is enough to take into account of the scaling properties (1.9) and use the fact that if $\Omega$ is the disjoint union of a family of open sets $\Omega _i$ with $i\in I$, then

$$\begin{aligned} T_p^{1/(p-1)}(\Omega )=\sum _{i\in I}T_p^{1/(p-1)}(\Omega _i),\quad \lambda _p(\Omega )=\inf _{i\in I}\lambda _p(\Omega _i). \end{aligned}$$

(1.13)

Then, choosing $\Omega _n$ as the disjoint union of n balls with measure 1/n each and taking the limit as $n\rightarrow \infty$, gives

$$\begin{aligned} T_p(\Omega _n)=\omega _d^{1-p-p/d}n^{-p/d}T_p(B_1)\rightarrow 0 \end{aligned}$$

and

$$\begin{aligned} \lambda _p(\Omega _n)=\omega ^{p/d}n^{p/d}\lambda _p(B_1)\rightarrow +\infty . \end{aligned}$$

Thus, a characterization of $\inf /\sup$ of the quantity $\lambda _p^\alpha (\Omega )T_p^\beta (\Omega )$, among the domains $\Omega \subset {\mathbb {R}}^d$ with unitary measure, when $\alpha =0$ or $\beta =0$ or $\alpha \beta <0$, follows by (1.10), (1.11) and (1.12).

It remains to consider the case $\alpha >0$ and $\beta >0$. Setting $q=\beta /\alpha >0$ we can limit ourselves to deal with the quantity

$$\begin{aligned} \lambda _p(\Omega )T_p^q(\Omega ). \end{aligned}$$

Using the scaling properties (1.9) we can remove the constraint of prescribed Lebesgue measure on $\Omega$ by normalizing the quantity $\lambda _p(\Omega )T_p^q(\Omega )$, multiplying it by a suitable power of $|\Omega |$. We then end up with the scaling invariant shape functional

$$\begin{aligned} F_{p,q}(\Omega )=\frac{\lambda _p(\Omega )T_p^q(\Omega )}{|\Omega |^{\alpha (p,q,d)}}\quad \text {with}\quad \alpha (p,q,d):=q(p-1)+\frac{p(q-1)}{d}\;, \end{aligned}$$

that we want to minimize or maximize over the class of open sets $\Omega \subset {\mathbb {R}}^d$ with $0<|\Omega |<\infty$.

The limit cases, when $p=1$ and $p=+\infty$, are also meaningful. When $p\rightarrow 1$ the quantities $\lambda _p(\Omega )$ and $T_p(\Omega )$ are related to the notion of Cheeger constant $h(\Omega )$, see definition (5.1). In particular we obtain as a natural “limit” functional

$$\begin{aligned} F_{1,q}(\Omega )=\left( h(\Omega )|\Omega |^{1/d}\right) ^{1-q} \end{aligned}$$

whose optimization problems are well studied in the literature. Concerning the case $p=+\infty$, we show that the family $F_{p,q}^{1/p}$ pointwise converges, as $p\rightarrow \infty$, to the shape functional

and we study the related optimization problems in the class of all domains $\Omega$ and in that of convex domains.

The study of the functionals $F_{p,q}$ has been already considered in the literature. The case when $p=2$ has been extensively discussed in [30,31,32,33] (see also [9]) and our results can be seen as natural extensions. An interesting variant, where the shape functionals involve the $L^\infty$ norm of the function $w_p$ solution of (1.2) has been considered in [19] in the case $p=2$.

The paper is organized as follows. In the first three sections we study the optimization problems for $F_{p,q}$, when $1<p<\infty$ and in different classes of domains. More precisely: in Sect. 2 we consider the class of all open sets of ${\mathbb {R}}^d$ with finite Lebesgue measure, in Sect. 3 we consider the class of bounded convex open sets and in Sect. 4 that of thin domains which will be suitable defined. The analysis of the optimization problems in the extremal cases (respectively when $p=1$ and $p=+\infty$) are contained in Sects. 5 and 6. Finally Sect. 7 contains a list of several open problems which we believe may be interest for future researches. For the sake of completeness we add an appendix section devoted to clarify the assumptions we use for the limit case of Sect. 6.

2 Optimization for general domains

The crucial inequality to provide a lower bound to $F_{p,q}$ is the Kohler-Jobin inequality, first proved for $p=2$ in [25, 26] and then for a general p in [7], which asserts that balls minimize principal frequency among all sets of prescribed torsional rigidity. More precisely we have

$$\begin{aligned} \lambda _p(B)T_p^{p'/(p'+d)}(B)\le \lambda _p(\Omega )T_p^{p'/(p'+d)}(\Omega ). \end{aligned}$$

(2.1)

Proposition 2.1

Let $1<p<+\infty$. Then

$$\begin{aligned} {\left\{ \begin{array}{ll} \min \big \{F_{p,q}(\Omega )\ :\ \Omega \text { open in }{\mathbb {R}}^d,\ 0<|\Omega |<\infty \big \}=F_{p,q}(B)&{}\hbox {if}\quad 0<q\le p'/(p'+d);\\ \inf \big \{F_{p,q}(\Omega )\ :\ \Omega \text { open in }{\mathbb {R}}^d,\ 0<|\Omega |<\infty \big \}=0\,&{}\hbox {if}\quad q>p'/(p'+d), \end{array}\right. } \end{aligned}$$

where B is any ball in ${\mathbb {R}}^d$.

Proof

We denote for the sake of brevity ${{\bar{q}}}=p'/(p'+d)$. Notice that

$$\begin{aligned} \alpha (p,q,d)=\frac{[d(p-1)+p][q-{{\bar{q}}}]}{d} \end{aligned}$$

and thus

$$\begin{aligned} F_{p,q}(\Omega )=\lambda (\Omega )T^{{{\bar{q}}}}(\Omega )\Big [\frac{T(\Omega )}{|\Omega |^{[d(p-1)+p]/d}}\Big ]^{q-{{\bar{q}}}}. \end{aligned}$$

By Kohler-Jobin inequality (2.1) and Saint-Venant inequality (1.10) we get the thesis for $0<q\le {{\bar{q}}}$. Now, let $\Omega$ be the disjoint union of $B_1$ and N disjoint balls of radius ${\varepsilon }\in (0,1]$. Taking into account (1.13) we have

$$\begin{aligned} F_{p,q}(\Omega )=F_{p,q}(B_1)\,\frac{(1+N{\varepsilon }^{d+p/(p-1)})^{q(p-1)}}{(1+N{\varepsilon }^d)^{(d(p-1)+p)(q-{{\bar{q}}})/d}}. \end{aligned}$$

Taking now $N{\varepsilon }^{d+p/(p-1)}=1$ gives

$$\begin{aligned} F_{p,q}(\Omega )\le F_{p,q}(B_1)\frac{2^{q(p-1)}}{(1+{\varepsilon }^{-p/(p-1)})^{(d(p-1)+p)(q-{{\bar{q}}})/d}}, \end{aligned}$$

which vanishes as ${\varepsilon }\rightarrow 0$ as soon as $q>{{\bar{q}}}$. $\square$

In dealing with the supremum of $F_{p,q}$ a natural threshold arises from the Polya inequality whose brief proof we recall.

Proposition 2.2

For every $\Omega \subset {\mathbb {R}}^d$ with $0<|\Omega |<+\infty$ and every $1<p<+\infty$ we have

$$\begin{aligned} F_{p,1}(\Omega )=\frac{\lambda _p(\Omega )T_p(\Omega )}{|\Omega |^{p-1}}\le 1. \end{aligned}$$

(2.2)

Proof

Let $w_\Omega$ be the solution to (1.2). By the definition of $\lambda _p(\Omega )$ and by Hölder inequality we have

$$\begin{aligned} \lambda _p(\Omega )\le \frac{\int _\Omega |\nabla w_p|^p dx}{\int _\Omega w_p^p dx}=\frac{\int _\Omega w_p dx}{\int _\Omega w_p^p dx}\le \frac{|\Omega |^{p-1}}{\left( \int _\Omega w_p dx \right) ^{p-1}}. \end{aligned}$$

The conclusion follows by (1.3). $\square$

Proposition 2.3

Let $1<p<\infty$. Then

$$\begin{aligned} {\left\{ \begin{array}{ll} \sup \big \{F_{p,q}(\Omega )\ :\ \Omega \text { open in }{\mathbb {R}}^d,\ 0<|\Omega |<\infty \big \}=+\infty &{}\hbox {if}\quad 0<q<1;\\ \sup \big \{F_{p,q}(\Omega )\ :\ \Omega \text { open in }{\mathbb {R}}^d,\ 0<|\Omega |<\infty \big \}\le T^{q-1}_p(B)/|B|^{(q-1)(p-1+p/d)}&{}\hbox {if}\quad q\ge 1. \end{array}\right. } \end{aligned}$$

Proof

Let $\Omega _N$ be the disjoint union of N balls of unitary radius. By (1.13) we have

$$\begin{aligned} F_{p,q}(\Omega _N)=N^{(1-q)(p-1+p/d)}F_{p,q}(B_1). \end{aligned}$$

Taking the limit as $N\rightarrow \infty$ we have $F_{p,q}(\Omega _N)\rightarrow +\infty$ whenever $0<q<1$. Moreover, when $q\ge 1$, using Proposition 2.2 and the Saint-Venant inequality (1.10), we have

$$\begin{aligned} F_{p,q}(\Omega )=F_{p,1}(\Omega )\Big (\frac{T_p(\Omega )}{|\Omega |^{p-1+p/d}}\Big )^{q-1}\le \Big (\frac{T_p(B)}{|B|^{p-1+p/d}}\Big )^{q-1}. \end{aligned}$$

which concludes the proof. $\square$

When $p=2$ and $q=1$ the upper bound given in the Proposition 2.2 is sharp as first proved in [32]. Using the theory of capacitary measures, a shorther proof was given in [30]. The latter extends, naturally, to the case when $p\le d$ and $q=1$ as we show in the proposition below.

Proposition 2.4

Let $1<p\le d$. Then

$$\begin{aligned} \sup \big \{F_{p,1}(\Omega )\ :\ \Omega \subset {\mathbb {R}}^d\,\hbox {open},\,0<|\Omega |<+\infty \big \}=1. \end{aligned}$$

Proof

By repeating the construction made in [13] (see also Remark 4.3.11 and Example 4.3.12 of [10], and references therein) we have that for every p-capacitary measure $\mu$ (that is a nonnegative Borel measure, possibly taking the value $+\infty$, with $\mathrm {cap}_p(E)=0\Longrightarrow \mu (E)=0$) there exists a sequence $(\Omega _n)$ of (smooth) domains such that

$$\begin{aligned} \lambda _p(\Omega _n)\rightarrow \lambda _p(\mu ),\quad T_p(\Omega _n)\rightarrow T_p(\mu ),\quad |\Omega _n|\rightarrow |\{\mu <+\infty \}|, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&\lambda _p(\mu )=\min \bigg \{\int |\nabla u|^p\,dx+\int |u|^p\,d\mu \ :\ u\in W^{1,p}({\mathbb {R}}^d)\cap L^p_\mu ,\ \int |u|^p\,dx=1\bigg \},\\&T_p(\mu )=\max \bigg \{\bigg [\int |u|\,dx\bigg ]^p\bigg [\int |\nabla u|^p\,dx+\int |u|^p\,d\mu \bigg ]^{-1}\ :\ u\in W^{1,p}({\mathbb {R}}^d)\cap L^p_\mu {\setminus }\{0\}\bigg \}. \end{aligned} \end{aligned}$$

Taking the ball $B_1$ and for every $c>0$, we have

$$\begin{aligned} \sup _\Omega \bigg [\frac{\lambda _p(\Omega )T_p(\Omega )}{|\Omega |^{p-1}}\bigg ]=\sup _\mu \bigg [\frac{\lambda _p(\mu )T_p(\mu )}{|\{\mu <+\infty \}|^{p-1}}\bigg ]\ge \sup _{c>0}\bigg [\frac{\lambda _p(\mu _c)T_p(\mu _c)}{|B_1|^{p-1}}\bigg ]. \end{aligned}$$

Clearly $\lambda _p(\mu _c)=c+\lambda _p(B_1)$. Now, consider for $\delta >0$ the function

$$\begin{aligned}u_\delta (x)= {\left\{ \begin{array}{ll} 1&{}\hbox {if}\quad |x|\le 1-\delta ,\\ (1-|x|)/\delta &{}\hbox {if}\quad |x|>1-\delta . \end{array}\right. } \end{aligned}$$

We have

$$\begin{aligned} \begin{aligned} T_p(\mu _c)&\ge \Big [\int _{B_1}u_\delta \,dx\Big ]^p\Big [\int _{B_1}|\nabla u_\delta |^p\,dx+c\int _{B_1}u_\delta ^p\,dx\Big ]^{-1}\\&\ge \Big [\omega _d(1-\delta )^d\Big ]^p\Big [\delta ^{-p}\omega _d\big (1-(1-\delta )^d\big )+c\omega _d\Big ]^{-1}\\&\ge \omega _d^{p-1}(1-\delta )^{pd}\Big [\delta ^{-p}+c\Big ]^{-1}. \end{aligned}\end{aligned}$$

Therefore

$$\begin{aligned} \frac{\lambda _p(\mu _c)T_p(\mu _c)}{|B_1|^{p-1}}\ge \frac{(c+\lambda _p(B_1))(1-\delta )^{pd}}{\delta ^{-p}+c}. \end{aligned}$$

By letting $c\rightarrow +\infty$ and then $\delta \downarrow 0$ we obtain the thesis. $\square$

3 Optimization in convex domains

We now deal with the optimization problems in the class of convex domains. Notice that adding in (1.11) and in (1.12) a convexity constraint on the admissible domains $\Omega$ does not change the values of $\inf$ and $\sup$. To see this one can take a unit measure normalization of the following convex domains (slab shape)

$$\begin{aligned} C_{A,{\varepsilon }}:=A\times (-{\varepsilon },{\varepsilon }) \end{aligned}$$

(3.1)

being A a convex $d-1$ dimensional open set with finite $d-1$ dimensional measure and use the following Lemma, which will be proved in a slightly more general version in Proposition 4.1 of Sect. 4.

Lemma 3.1

Let $A\subset {\mathbb {R}}^{d-1}$ be a bounded open set and let ${\varepsilon }>0$. Let $C_{A,{\varepsilon }}:=A\times (-{\varepsilon },{\varepsilon })$. Then we have

$$\begin{aligned} T_p(C_{A,{\varepsilon }})\le \left( {\mathcal {H}}^{d-1}(A)\right) ^{p-1}{\varepsilon }^{2p-1}\left( \frac{2}{p'+1}\right) ^{p-1},\quad \lambda _p(C_{A,{\varepsilon }})\ge {\varepsilon }^{-p}\left( \frac{\pi _p}{2}\right) ^p, \end{aligned}$$

where $\pi _p$ is given in (1.7). In addition, as ${\varepsilon }\rightarrow 0$, we have

$$\begin{aligned} T_p(C_{A,{\varepsilon }})\approx \left( {\mathcal {H}}^{d-1}(A)\right) ^{p-1}{\varepsilon }^{2p-1}\left( \frac{2}{p'+1}\right) ^{p-1}, \quad \lambda _p(C_{A,{\varepsilon }})\approx {\varepsilon }^{-p}\left( \frac{\pi _p}{2}\right) ^p. \end{aligned}$$

By using the previous lemma we have also

$$\begin{aligned} \lim _{{\varepsilon }\rightarrow 0}F_{p,q}(C_{A,{\varepsilon }})= {\left\{ \begin{array}{ll} 0&{}\hbox {if}\quad q>1,\\ +\infty &{}\hbox {if}\quad q<1. \end{array}\right. } \end{aligned}$$

Hence the only interesting optimization problems in the class of convex domains are the following ones

$$\begin{aligned} \inf \{F_{p,q}(\Omega )\ :\ \Omega \subset {\mathbb {R}}^d\,\hbox {open, convex, bounded}\},\quad \hbox {with}\quad q\le 1, \\ \sup \{F_{p,q}(\Omega )\ :\ \Omega \subset {\mathbb {R}}^d\,\hbox {open, convex, bounded}\},\quad \hbox {with}\quad q\ge 1. \end{aligned}$$

We denote respectively by $m_{p,q}$ and $M_{p,q}$ the two quantities above.

With the convexity constraint, the so called Hersch–Protter inequality holds (for a proof see for instance [8, 22]):

$$\begin{aligned} \lambda _p(\Omega )\ge \left( \frac{\pi _p}{2\rho (\Omega )}\right) ^p. \end{aligned}$$

(3.2)

Moreover, the p-torsional rigidity of a bounded convex open set satisfies the following generalization of Makai inequality (see [28, Theorem 4.3]):

$$\begin{aligned} \frac{T_p(\Omega )}{|\Omega |^{p-1}}\le \frac{\rho ^p(\Omega )}{(p'+1)^{p-1}}. \end{aligned}$$

(3.3)

Both inequalities are sharp and the equality is asymptotically attained by taking, for instance, the sequence $C_{A,{\varepsilon }}$ of Lemma 3.1. Taking advantage of (3.2) and (3.3) we can show the following bounds.

Proposition 3.2

Let $1<p<+\infty$. Then

$$\begin{aligned}&m_{p,1}\ge \frac{1}{d}\Big (\frac{\pi _p}{2}\Big )^p\Big (\frac{1}{d+p'}\Big )^{p-1},\end{aligned}$$

(3.4)

$$\begin{aligned}&M_{p,1}\le \min \bigg \{1,\Big (\frac{1}{p'+1}\Big )^{p-1} \lambda _p(B_1)\bigg \}. \end{aligned}$$

(3.5)

Proof

Let $\Omega \subset {\mathbb {R}}^{d}$ be any bounded convex set. Without loss of generality, we can suppose $0\in \Omega$. We denote by $j_{\Omega }(x)$ the Minkowski functional (also known as gauge function) of $\Omega$, that is

$$\begin{aligned} j_{\Omega }(x):=\inf \left\{ r>0: x\in r\Omega \right\} . \end{aligned}$$

The main properties of $j_{\Omega }$ are summarized in Lemma 2.3 of [9]. In particular we recall that $j_\Omega$ is a convex, Lipschitz, 1-positively homogeneous function, ${\mathcal {H}}^{d-1}$-a.e. differentiable in $\partial \Omega$, and satisfies

$$\begin{aligned} |\nabla j_{\Omega }(x)|^{-1}=x\cdot \nu _{\Omega }(x), \quad \text {for}\quad {\mathcal {H}}^{d-1}\text {-a.e. }x\in \partial \Omega , \end{aligned}$$

(3.6)

being $\nu _{\Omega }(x)$ the outer normal unit versor at the point $x\in \partial \Omega$. We consider

$$\begin{aligned} u(x):=1-j^{p'}_\Omega (x)\in W^{1,p}_0(\Omega ). \end{aligned}$$

By using coarea formula (3.6) and the divergence theorem it is easy to prove that

$$\begin{aligned} \int _\Omega u(x) dx=|\Omega |-\int _0^{1}t^{p'+d-1}dt\int _{\partial \Omega }|j_{\Omega }(x)|^{-1}d{\mathcal {H}}^{d-1}(x)=\frac{p'}{d+p'}|\Omega |, \end{aligned}$$

and

$$\begin{aligned} \int _\Omega |\nabla u(x)|^{p}dx=\frac{p'^p}{d+p'}\int _{\partial \Omega }|\nabla j_{\Omega }(x)|^{p-1}d{\mathcal {H}}^{d-1}(x)\le \frac{dp'^{p}}{d+p'}|\Omega |\rho ^{-p}(\Omega ), \end{aligned}$$

where the last inequality follows by the fact that

$$\begin{aligned} \rho (\Omega )\le x\cdot \nu _{\Omega }(x), \text {for}\quad {\mathcal {H}}^{d-1}\text {-a.e. }x\in \partial \Omega , \end{aligned}$$

see Lemma 2.1 in [9]. Hence by testing (1.1) with the function u we have

$$\begin{aligned} \frac{T_p(\Omega )}{\rho (\Omega )^p|\Omega |^{p-1}}\ge \frac{1}{d}\big (\frac{1}{ d+p'}\big )^{p-1}. \end{aligned}$$

(3.7)

Taking into account (3.2), we obtain

$$\begin{aligned} F_{p,1}(\Omega )\ge \frac{1}{d}\big (\frac{1}{ d+p'}\big )^{p-1}\lambda _p(\Omega )\rho (\Omega )^p\ge \frac{1}{d} \Big (\frac{\pi _p}{2}\Big )^p \big (\frac{1}{d+p'}\big )^{p-1}, \end{aligned}$$

which proves (3.4).

To prove the second inequality we use (3.3) and the inequality

$$\begin{aligned} \lambda _p(\Omega )\le \lambda _p(B_1)\rho (\Omega )^{-p}, \end{aligned}$$

to obtain

$$\begin{aligned} F_{p,1}(\Omega )\le \Big (\frac{1}{p'+1}\Big )^{p-1}\rho ^p(\Omega )\lambda _p(\Omega )\le \Big (\frac{1}{p'+1}\Big )^{p-1}\lambda _p(B_1), \end{aligned}$$

which, together with Proposition 2.2, gives (3.5). $\square$

Remark 3.3

We stress here that inequality (3.7) has been already proved in [12, 28]. However, their results are given in the more general anisotropic setting where the proofs become more involved.

Remark 3.4

Combining inequalities (3.5) and (1.8), we obtain

$$\begin{aligned} F_{p,1}(\Omega )\le \Big (\frac{1}{p'+1}\Big )^{p-1}\frac{(p+1)(p+2)\cdots (p+d)}{d!}\le \frac{(1+p)^d}{2^{p-1}}. \end{aligned}$$

Thereby, as soon as p is large enough, we have $M_{p,1}<1$.

When $q\ne 1$ the values $m_{p,q}$ and $M_{p,q}$ are achieved by some optimal domains, as shown in the next theorem.

Theorem 3.5

Let $1<p<+\infty$. Then

$$\begin{aligned} {\left\{ \begin{array}{ll} m_{p,q}\ge m_{p,1}T^{q-1}_p(B)/|B|^{(d(p-1)+p)(q-1)/d}&{}\hbox {if}\quad q<1,\\ M_{p,q}\le M_{p,1}T^{q-1}_p(B)/|B|^{(d(p-1)+p)(q-1)/d}&{}\hbox {if}\quad q>1. \end{array}\right. } \end{aligned}$$

Moreover, there exist convex domains $\Omega ^m_{p,q}$ and $\Omega ^M_{p,q}$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} F_{p,q}(\Omega ^m_{p,q})=m_{p,q}&{}\hbox {if}\quad q<1,\\ F_{p,q}(\Omega ^M_{p,q})=M_{p,q}&{}\hbox {if}\quad q>1. \end{array}\right. } \end{aligned}$$

Proof

The first part follows at once using Saint-Venant inequality (1.10) together with the equality

$$\begin{aligned} F_{p,q}(\Omega )=F_{p,1}(\Omega )\Big (\frac{T_p(\Omega )}{|\Omega |^{p-1+p/d}}\Big )^{q-1}. \end{aligned}$$

Concerning the existence of optimal convex domains, we can repeat the argument used in [30]. First we notice that

$$\begin{aligned} F_{p,q}(\Omega )=\frac{F^q_{p,1}(\Omega )\lambda _p(\Omega )^{1-q}}{|\Omega |^{p(q-1)/d}}. \end{aligned}$$

(3.8)

Moreover, any convex open set $\Omega$ contains a two-sided cone with base area equal to a $d-1$ dimensional disk of radius $\rho (\Omega )$ and total height equal to ${{\,\mathrm{diam}\,}}(\Omega )$, hence

$$\begin{aligned} |\Omega |\ge d^{-1}\omega _{d-1}{{\,\mathrm{diam}\,}}(\Omega )\rho (\Omega )^{d-1}. \end{aligned}$$

(3.9)

Thus, suppose $0<q<1$ and let $(\Omega _n)$ be a minimizing sequence for $F_{p,q}$ made up of convex domains. By scaling invariance we can suppose $\rho (\Omega _n)=1$. For n large enough we have $F_{p,q}(\Omega _n)\le F_{p,q}(B)$. Using (3.2) and (3.8) we have

$$\begin{aligned} \frac{F_{p,q}(B)}{m^q_{p,1}}\ge \left( \frac{\pi _p}{2}\right) ^{p(1-q)}|\Omega _n|^{p(1-q)/d}. \end{aligned}$$

Combining the last estimate with (3.9) we have

$$\begin{aligned} \sup _n{{\,\mathrm{diam}\,}}(\Omega _n)<+\infty . \end{aligned}$$

Hence, up to translations, the whole sequence $(\Omega _n)$ is contained in a compact set and we can extract a subsequence $(\Omega _{n_k})$ which converges in both Hausdorff and co-Hausdorff distance to some $\Omega ^m_{p,q}$ (see [17], for details about these convergences). Using the well-known continuity properties for $\lambda _p$, $T_p$ and Lebesgue measure with respect to Hausdorff metrics on the class of bounded convex sets, we conclude that

$$\begin{aligned} m_{p,q}=\lim _{n\rightarrow \infty }F_{p,q}(\Omega _n)=F_{p,q}(\Omega ^m_{p,q}). \end{aligned}$$

If $q>1$ we can follow the similar strategy and consider a maximizing sequence $(\Omega _n)$ with unitary inradius. By (3.8) and (3.2) we have, for n large enough,

$$\begin{aligned} F_{p,q}(B)\le F_{p,q}(\Omega _n)\le M^q_{p,1}\left( \frac{\pi _p}{2}\right) ^{p(1-q)}\left( \frac{1}{|\Omega _n|^{1/d}}\right) ^{p(q-1)}, \end{aligned}$$

which, thanks to (3.9), implies again $\sup _n{{\,\mathrm{diam}\,}}(\Omega _n)<+\infty$. $\square$

4 Optimization for thin domains

In this section we study the optimization problems for the functionals $F_{p,1}$ in the class of the so-called thin domains, which has been already considered in [30] for $p=2$. By a thin domain we mean a family of open sets $(\Omega _{\varepsilon })_{{\varepsilon }>0}$, of the form

$$\begin{aligned} \Omega _{\varepsilon }:=\big \{(x,y)\in A\times {\mathbb {R}}\ :\ {\varepsilon }h_-(x)<y<{\varepsilon }h_+(x)\big \}\; , \end{aligned}$$

(4.1)

where A is $(d-1)$-dimensional open set, $h_-,h_+$ are real bounded measurable functions defined on A and ${\varepsilon }$ is a small parameter. We assume $h_+\ge h_-$ and we denote by h(x) the local thickness function

$$\begin{aligned} h(x)=h_+(x)-h_-(x)>0\quad \text {on}\quad A. \end{aligned}$$

Moreover we say that the thin domain $(\Omega _{\varepsilon })_{{\varepsilon }>0}$ is convex if the corresponding domain A is convex and the local thickness function h is concave. The volume of $\Omega _{\varepsilon }$ is clearly given by

$$\begin{aligned} |\Omega _{\varepsilon }|={\varepsilon }\int _A h\,dx, \end{aligned}$$

while we can compute the behaviour (as ${\varepsilon }\rightarrow 0$) of $T_p(\Omega _{\varepsilon })$ and $\lambda _p(\Omega _{\varepsilon })$ by means of the following proposition (in the case $p=2$ a more refined asymptotics can be found in [5, 6]). From now on, we write the norms $\Vert \cdot \Vert _{p}$, omitting the dependence on the domain.

Proposition 4.1

Let $A\subset {\mathbb {R}}^{d-1}$ be an open set with finite ${\mathcal {H}}^{d-1}$-measure and $h_-,h_+\in C^{1}(A)$ with $h_+>h_-$. Let $\Omega _{\varepsilon }$ be defined by (4.1). We have

$$\begin{aligned} T_p(\Omega _{\varepsilon })\le \frac{{\varepsilon }^{2p-1}}{2^p}\left( \frac{1}{p'+1}\right) ^{p-1}\left( \int _A h^{p'+1}dx\right) ^{p-1},\quad \lambda _p(\Omega _{\varepsilon })\ge {\varepsilon }^{-p}\left( \frac{\pi _p}{\Vert h\Vert _\infty }\right) ^p,\nonumber \\ \end{aligned}$$

(4.2)

where $\pi _p$ is given in (1.7). In addition, as ${\varepsilon }\rightarrow 0$, we have

$$\begin{aligned} T_p(\Omega _{\varepsilon })\approx \frac{{\varepsilon }^{2p-1}}{2^p}\left( \frac{1}{p'+1}\right) ^{p-1}\left( \int _A h^{p'+1}dx\right) ^{p-1},\quad \lambda _p(\Omega _{\varepsilon })\approx {\varepsilon }^{-p}\left( \frac{\pi _p}{\Vert h\Vert _\infty }\right) ^p.\nonumber \\ \end{aligned}$$

(4.3)

Proof

First we deal with inequalities (4.2). Let $\phi \in C^{\infty }_c(\Omega _{\varepsilon })$; since the function $\phi (x,\cdot )$ is admissible to compute $T_p\left( {\varepsilon }h_-(x),{\varepsilon }h_+(x)\right)$, by (1.1) we obtain

$$\begin{aligned} \begin{aligned} \int _{{\varepsilon }h_-(x)}^{{\varepsilon }h_+(x)} \phi (x,\cdot )dy&\le T^{1/p}_p({\varepsilon }h_-(x),{\varepsilon }h_+(x))\left( \int ^{{\varepsilon }h_+(x)}_{{\varepsilon }h_-(x)}|\nabla _y\phi (x,\cdot )|^p dy\right) ^{1/p}\\&\le T^{1/p}_p({\varepsilon }h_-(x),{\varepsilon }h_+(x))\left( \int ^{{\varepsilon }h_+(x)}_{{\varepsilon }h_-(x)}|\nabla \phi (x,\cdot )|^p dy\right) ^{1/p}. \end{aligned} \end{aligned}$$

(4.4)

Taking into account (1.5) we have

$$\begin{aligned} T_p\left( {\varepsilon }h_-(x),{\varepsilon }h_+(x)\right) =\frac{{\varepsilon }^{2p-1}}{2^p}\left( \frac{p-1}{2p-1}\right) ^{p-1}h^{2p-1}(x), \end{aligned}$$

(4.5)

and thus, integrating on A in (4.4), we deduce

$$\begin{aligned} \left( \int _{\Omega _{\varepsilon }}\phi (x,y)\,dxdy\right) ^p\le & {} \frac{{\varepsilon }^{2p-1}}{2^p}\left( \frac{p-1}{2p-1}\right) ^{p-1}\\&\bigg [\int _A h^{(2p-1)/p}\bigg (\int ^{{\varepsilon }h_+(x)}_{{\varepsilon }h_-(x)}|\nabla \phi (x,\cdot )|^p\,dy\bigg )^{1/p}dx\bigg ]^p. \end{aligned}$$

Hölder inequality now gives

$$\begin{aligned} \left( \int _{\Omega _{\varepsilon }}\phi (x,y)\,dxdy\right) ^p\le & {} \frac{{\varepsilon }^{2p-1}}{2^p}\left( \frac{p-1}{2p-1}\right) ^{p-1}\left( \int _A h^{(2p-1)/(p-1)}dx\right) ^{p-1}\\&\int _{\Omega _{\varepsilon }}|\nabla \phi (x,y)|^pdxdy. \end{aligned}$$

Since $\phi$ is arbitrary and $p'+1=(2p-1)/(p-1)$, we conclude that

$$\begin{aligned} T_p(\Omega _{\varepsilon })\le \frac{{\varepsilon }^{2p-1}}{2^p}\left( \frac{1}{p'+1}\right) ^{p-1}\left( \int _A h^{p'+1}dx\right) ^{p-1}. \end{aligned}$$

To get the second inequality in (4.2) we notice that, by (1.6), for every $\phi \in C^\infty _c(\Omega _{\varepsilon })$ we have

$$\begin{aligned} \lambda _p({\varepsilon }h_-(x),{\varepsilon }h_+(x))\int _{{\varepsilon }h_-(x)}^{{\varepsilon }h_+(x)}|\phi (x,\cdot )|^{p} dy\le & {} \int _{{\varepsilon }h_-(x)}^{{\varepsilon }h_+(x)}|\nabla _y \phi (x,\cdot )|^p dy \\\le & {} \int _{{\varepsilon }h_-(x)}^{{\varepsilon }h_+(x)}|\nabla \phi (x,\cdot )|^p dy. \end{aligned}$$

Since

$$\begin{aligned} \lambda _p({\varepsilon }h_-(x),{\varepsilon }h_+(x))= h^{-p}(x){\varepsilon }^{-p}\pi _p^p\ge \Vert h\Vert _{\infty }^{-p}{\varepsilon }^{-p}\pi _p^p, \end{aligned}$$

integrating on A and minimizing on $\phi$, we obtain

$$\begin{aligned} \lambda _p(\Omega _{\varepsilon })\ge \Vert h\Vert _{\infty }^{-p}{\varepsilon }^{-p}\pi _p^p. \end{aligned}$$

We now prove (4.3) for $T_p(\Omega _{\varepsilon })$. To this end we consider the function

$$\begin{aligned} w_{\varepsilon }(x,y):={\varepsilon }^{p'} h^{p'}(x)w\left( \frac{y-{\varepsilon }h_-(x)}{{\varepsilon }h(x)}\right) , \end{aligned}$$

where w denotes the solution to (1.2) when $\Omega =(0,1)$ and $d=1$ (for the sake of brevity we omit the dependence on p). Notice that $w_{\varepsilon }(x,\cdot )$ solves (1.2) in the interval $({\varepsilon }h_-(x),{\varepsilon }h_+(x))$. In particular, by using (1.3) and (4.5), we have

$$\begin{aligned} \int _{{\varepsilon }h_-(x)}^{{\varepsilon }h_+(x)}w_{\varepsilon }(x,y)dy=\int _{{\varepsilon }h_-(x)}^{{\varepsilon }h_+(x)}|\nabla _y w_{\varepsilon }(x,y)|^pdy= \frac{{\varepsilon }^{p'+1}h^{p'+1}(x)}{2^{p/(p-1)}(p'+1)}. \end{aligned}$$

A simple computation shows that

$$\begin{aligned} \nabla _y w_{\varepsilon }(x,y)={\varepsilon }^{p'-1}W_1(x,y),\quad \nabla _x w_{\varepsilon }(x,y)=-{\varepsilon }^{p'-1}W_1(x,y)\frac{y\nabla h(x)}{h(x)}+{\varepsilon }^{p'} W_2(x,y), \end{aligned}$$

where

$$\begin{aligned} W_1(x,y)=h^{p'-1}(x)w'\left( \frac{y-{\varepsilon }h_-(x)}{{\varepsilon }h(x)}\right) , \end{aligned}$$

and

$$\begin{aligned} W_2(x,y)=p'h^{p'-1}(x)\nabla h(x) w\left( \frac{y-{\varepsilon }h_-(x)}{{\varepsilon }h(x)}\right) -h^{p'}(x)w'\left( \frac{y-{\varepsilon }h_-(x)}{{\varepsilon }h(x)}\right) \nabla \left( \frac{h_-(x)}{h(x)}\right) . \end{aligned}$$

In particular

$$\begin{aligned} \begin{aligned}&\int _{{\varepsilon }h_-(x)}^{{\varepsilon }h_+(x)}|\nabla w_{\varepsilon }(x,y)|^pdy\\&\quad =\int _{{\varepsilon }h_-(x)}^{{\varepsilon }h_+(x)}\left\{ \left| {\varepsilon }^{p'-1}W_1(x,y)\right| ^{2}+\left| -{\varepsilon }^{p'-1}W_1(x,y)\frac{y\nabla h(x)}{h(x)}+{\varepsilon }^{p'}W_2(x,y)\right| ^2\right\} ^{p/2}dy. \end{aligned} \end{aligned}$$

By exploiting the change of variable $z=\frac{y-{\varepsilon }h_-(x)}{{\varepsilon }h(x)}$ in the latter identity, we conclude that, as ${\varepsilon }\rightarrow 0$,

$$\begin{aligned} \int _{{\varepsilon }h_-(x)}^{{\varepsilon }h_+(x)}|\nabla w_{\varepsilon }(x,y)|^p\,dy\approx \int _{{\varepsilon }h_-(x)}^{{\varepsilon }h_+(x)}|\nabla _y w_{\varepsilon }(x,y)|^pdy. \end{aligned}$$

Let $\phi \in C^\infty _c(A)$. Since the function $v(x,y)=\phi (x)w_{\varepsilon }(x,y)$ is admissible in (1.1), we get

$$\begin{aligned} T(\Omega _{\varepsilon })\ge \left( \int _{\Omega _{\varepsilon }}w_{\varepsilon }(x,y)\phi (x)\,dxdy\right) ^p\left( \int _{\Omega _{\varepsilon }}\left| \nabla \left( w_{\varepsilon }(x,y)\phi (x)\right) \right| ^p\,dxdy\right) ^{-1}. \end{aligned}$$

(4.6)

Moreover, by using basically the same argument as above, we have also that

$$\begin{aligned} \int _{\Omega _{\varepsilon }} \left| \nabla \left( w_{\varepsilon }(x,y)\phi (x)\right) \right| ^p\,dxdy\approx \int _{\Omega _{\varepsilon }} \left| \nabla _y w_{\varepsilon }(x,y)\right| ^p\left| \phi (x)\right| ^pdxdy,\quad \hbox {as}\quad {\varepsilon }\rightarrow 0.\nonumber \\ \end{aligned}$$

(4.7)

By combining (4.6) and (4.7) we obtain

$$\begin{aligned} \lim _{{\varepsilon }\rightarrow 0} \frac{T_p(\Omega _{\varepsilon })}{{\varepsilon }^{2p-1}}\ge & {} \lim _{{\varepsilon }\rightarrow 0}\frac{1}{{\varepsilon }^{2p-1}} \left( \int _{\Omega _{\varepsilon }}w_{\varepsilon }(x,y)\phi (x)\,dxdy\right) ^p\\&\left( \int _{\Omega _{\varepsilon }}\left| \nabla _y w_{\varepsilon }(x,y)\right| ^p\left| \phi (x)\right| ^p\,dxdy\right) ^{-1}. \end{aligned}$$

Finally, by taking $\phi$ which approximates $1_A$ in $L^p(A)$ in the right hand side of the inequality above, we conclude that

$$\begin{aligned} \lim _{{\varepsilon }\rightarrow 0}\frac{T_p(\Omega _{\varepsilon })}{{\varepsilon }^{2p-1}}\ge \frac{1}{2^p}\left( \frac{1}{p'+1}\right) ^{p-1}\left( \int _A h^{p'+1}dx\right) ^{p-1}, \end{aligned}$$

and the thesis is achieved taking into account (4.2). The asymptotics in (4.3) for $\lambda _p$ can be treated with similar arguments. $\square$

Actually, by means of a density argument, we can drop the regularity assumptions on $h_+$ and $h_-$ and extend the formulas (4.2) and (4.3) to any family $(\Omega _{\varepsilon })_{{\varepsilon }>0}$ defined as in (4.1), with $h_+$ and $h_-$ bounded and measurable functions. We thus have:

$$\begin{aligned} F_{p,1}(\Omega _{\varepsilon })=\frac{\lambda _p(\Omega _{\varepsilon })T_p(\Omega _{\varepsilon })}{|\Omega _{\varepsilon }|^{p-1}}\approx \gamma _p\left( \Vert h\Vert ^{p'}_\infty \int _A h\,dx\right) ^{1-p}\left( \int _A h^{1+p'}dx\right) ^{p-1}, \end{aligned}$$

where

$$\begin{aligned} \gamma _p=\left( \frac{\pi _p }{2}\right) ^p\left( \frac{1}{p'+1}\right) ^{p-1}. \end{aligned}$$

We then define the functional $F_{p,1}$ on the thin domain $(\Omega _{\varepsilon })_{{\varepsilon }>0}$ associated with the $d-1$ dimensional domain A and the local thickness function h by

$$\begin{aligned} F_{p,1}(A,h)=\gamma _p\left( \frac{\int _A h^{p'+1}dx}{\Vert h\Vert ^{p'}_\infty \int _A h\,dx}\right) ^{p-1}. \end{aligned}$$

(4.8)

Our next goal is to give a complete solution to the optimization problems for the functional $F_{p,1}$ in the class of convex thin domains. To this aim we recall the following result (see Theorem 6.2 in [4]).

Theorem 4.2

Let $E\subset {\mathbb {R}}^N$ be a bounded open convex set, such that $0\in E$ and let $1\le s<r<\infty$. Then for every continuous function $h:E\rightarrow {\mathbb {R}}^+$ satisfying

$$\begin{aligned} h(\lambda x)\ge \lambda h(x)+(1-\lambda )\quad \forall x\in E,\ \forall \lambda \in (0,1), \end{aligned}$$

(4.9)

and such that $\Vert h\Vert _{L^\infty (E)}=1$, it holds

$$\begin{aligned} \int _E h^r(x)\,dx\ge C_{r,s}\int _E h^s(x)\,dx \end{aligned}$$

where

$$\begin{aligned} C_{r,s}=\frac{\int _0^1 (1-t)^{N-1}t^r\,dt}{\int _0^1 (1-t)^{N-1}t^s\,dt}. \end{aligned}$$

In addition, equality occurs if E is a ball of radius 1 and $h(x)=1-|x|$.

As an application we obtain the following lemma, which generalizes Proposition 5.2 in [30].

Lemma 4.3

Let $E\subset {\mathbb {R}}^N$ be a bounded open convex set and let $1<r<\infty$. Then for every concave function $h:E\rightarrow {\mathbb {R}}^+$ with $\Vert h\Vert _{L^\infty (E)}=1$ we have

$$\begin{aligned} \frac{\int _E h^r(x)\,dx}{\int _E h(x)\,dx}\ge (N+1)\left( {\begin{array}{c}N+r\\ N\end{array}}\right) ^{-1}. \end{aligned}$$

(4.10)

In addition, the inequality above becomes an equality when E is a ball of radius 1 and $h(x)=1-|x|$.

Proof

First we assume that $E\subset {\mathbb {R}}^N$ is a ball centered in the origin and h is a radially symmetric, decreasing, concave function $h:E\rightarrow [0,1]$ with $h(0)=1$. Then h satisfies (4.9) and we can apply Theorem 4.2 with $s=1$, to get

$$\begin{aligned} \int _E h^{r}(x)\,dx\ge C_{r,1}\int _E h(x)\,dx, \end{aligned}$$

where

$$\begin{aligned} C_{r,1}=\frac{\int _0^1 (1-t)^{N-1}t^{r}dt }{\int _0^1 (1-t)^{N-1}t\,dt}=\frac{\left( {\begin{array}{c}N+r\\ N\end{array}}\right) ^{-1}}{\left( {\begin{array}{c}N+1\\ N\end{array}}\right) ^{-1}}=(N+1)\left( {\begin{array}{c}N+r\\ N\end{array}}\right) ^{-1}. \end{aligned}$$

In order to get the inequality (4.10) in the general case, let $h^*:B\rightarrow [0,1]$ be the radially symmetric decreasing rearrangement of h, defined on the ball B centered at the origin and with the same volume as E. The standard properties of the rearrangement imply that

$$\begin{aligned} \int _B (h^*)^{r}(x)\,dx=\int _E h^{r}(x)\,dx,\quad \int _B h^*(x)\,dx=\int _E h(x)\,dx. \end{aligned}$$

Moreover, it is well-known that $h^*$ is concave. Since $h^*$ satisfies all the assumptions of the previous case, we get that $h^*$ (hence h) satisfies  (4.10). Finally, it is easy to show that the inequality in (4.10) holds as an equality for every cone function $h(x)=1-|x|$. $\square$

We are now in a position to show the main theorem of this section.

Theorem 4.4

Let $1<p<\infty$. Then

$$\begin{aligned} {\left\{ \begin{array}{ll} \sup \{F_{p,1}(A,h)\ :\ {\mathcal {H}}^{d-1}(A)<+\infty ,\ h\ge 0\}=\gamma _p\\ \inf \{F_{p,1}(A,h)\ :\ A\hbox { convex bounded,}\ h\ge 0, h\ \hbox {concave}\}= \gamma _pd^{p-1}\left( {\begin{array}{c}d+p'\\ d-1\end{array}}\right) ^{1-p}. \end{array}\right. } \end{aligned}$$

In addition, the first equality is attained taking h(x) to be any constant function while the second equality is attained taking as A the unit ball and as the local thickness function h(x) the function $1-|x|$.

Proof

Using definition (4.8) it is straightforward to prove that

$$\begin{aligned} F_{p,1}(A,h)\le \gamma _p \end{aligned}$$

and to verify that, if h is constant, then

$$\begin{aligned} F_{p,1}(A,h)=\gamma _p. \end{aligned}$$

Finally, by applying Lemma 4.3 with $N=d-1$, $E=A$ and $r=p'+1$ we obtain the second part of the theorem. $\square$

5 The case $p=1$

Given an open set $\Omega \subset {\mathbb {R}}^d$ with finite measure we define its Cheeger constant $h(\Omega )$ as

$$\begin{aligned} h(\Omega )=\inf \bigg \{\frac{P(E)}{|E|}\ :\ |E|>0,\ E\Subset \Omega \bigg \}, \end{aligned}$$

(5.1)

where $E\Subset \Omega$ means that ${\bar{E}}\subset \Omega$. Notice that in definition (5.1), thanks to a well-known approximation argument, we can evaluate the quotient P(E)/|E| among smooth sets which are compactly contained in $\Omega$. Following [24] we have

$$\begin{aligned} \lim _{p\rightarrow 1}\lambda _p(\Omega )=h(\Omega ), \end{aligned}$$

(5.2)

for every open set $\Omega$ with finite measure.

Remark 5.1

A caveat is necessary at this point: the usual definition of Cheeger constant as

$$\begin{aligned} c(\Omega )=\inf \bigg \{\frac{P(E)}{|E|}\ :\ |E|>0,\ E\subset \Omega \bigg \} \end{aligned}$$

is not appropriate to provide the limit equality (5.2), which would hold only assuming a mild regularity on $\Omega$ (for instance, it is enough to consider $\Omega$ which coincides with its essential interior, see [27]). To prove that in general $h(\Omega )\ne c(\Omega )$, one can consider $\Omega =B_1{\setminus }\partial B_{1/2}$. Then $c(\Omega )=c(B_1)=d$, while $h(\Omega )=2d$. The latter follows from the fact that, if $E\subset \Omega$, then $E=E_1\cup E_2$ where $E_1\Subset B_1{\setminus } {{\bar{B}}}_{1/2}$ and $E_2\Subset B_{1/2}$, together with the equality

$$\begin{aligned} h(B_1{\setminus } {{\bar{B}}}_{1/2})=h(B_{1/2})=2d. \end{aligned}$$

By the same argument used in [24] to prove (5.2) we can show that $T_p(\Omega )\rightarrow h^{-1}(\Omega )$ as $p\rightarrow 1$. For the sake of completeness we give the short proof below (see also Theorem 2 in [11]).

Proposition 5.2

Let $\Omega \subset {\mathbb {R}}^d$ be an open set with finite measure. Then, as $p\rightarrow 1$,

$$\begin{aligned} T_p(\Omega )\rightarrow h^{-1}(\Omega )\;. \end{aligned}$$

(5.3)

Proof

First we notice that for any $u\in C^{\infty }_c(\Omega )$, it holds:

$$\begin{aligned} \frac{\int _\Omega |\nabla u(x)|dx}{\int _\Omega |u(x)| dx}\ge h(\Omega ). \end{aligned}$$

(5.4)

Indeed, by assuming without loss of generality that $u\ge 0$, by coarea formula and Cavalieri’s principle, we have that

$$\begin{aligned} \int _\Omega |\nabla u|\,dx=\int _0^{+\infty }\!\!\!{\mathcal {H}}^{d-1}(\{u=t\})\,dt,\quad \int _\Omega u\,dx=\int _0^{+\infty }\!\!\!|\{u>t\}|\,dt. \end{aligned}$$

Since the sets $\{u>t\}\Subset \Omega$ are smooth for a.e. $t\in u(\Omega )$, (5.4) follows straightforwardly from (5.1). By combining (1.1) with (5.4) and Hölder inequality we then have

$$\begin{aligned} |\Omega |^{1-p}T_p(\Omega )\le h^{-p}(\Omega ), \end{aligned}$$

(5.5)

for any $1<p<\infty$.

Now, let $E_k\Subset \Omega$ be a sequence of smooth sets of $\Omega$ such that $P(E_k)/|E_k|\rightarrow h(\Omega )$. For a fixed k and any ${\varepsilon }>0$ small enough, we can find a Lipschitz function v compactly supported in $\Omega$, such that,

$$\begin{aligned} \chi _{E_k}\le v\le \chi _{E_{k,{\varepsilon }}},\quad |\nabla v|\le 1/{\varepsilon }\quad \hbox {in}\quad E_{k,{\varepsilon }}{\setminus } E_k, \end{aligned}$$

where $E_{k,{\varepsilon }}=E_k+B_{\varepsilon }$. Hence, by (1.1), we have

$$\begin{aligned} T_p(\Omega )\ge \frac{{\varepsilon }^{p}|E_k|^{p}}{|E_{k,{\varepsilon }}{\setminus } E_k|}. \end{aligned}$$

By first passing to the limit as $p\rightarrow 1$, and then as ${\varepsilon }\rightarrow 0$ we get

$$\begin{aligned} \liminf _{p\rightarrow 1}T_p(\Omega )\ge \frac{|E_k|}{P(E_k)}, \end{aligned}$$

which implies, as $k\rightarrow \infty$,

$$\begin{aligned} \liminf _{p\rightarrow 1}T_p(\Omega )\ge h^{-1}(\Omega ). \end{aligned}$$

Finally we conclude, taking into account (5.5). $\square$

The limits (5.2) and (5.3) justify the following definition:

$$\begin{aligned} F_{1,q}(\Omega ):=\left( h(\Omega )|\Omega |^{1/d}\right) ^{1-q}. \end{aligned}$$

Notice that $F_{p,q}(\Omega )\rightarrow F_{1,q}(\Omega )$ as $p\rightarrow 1$. In the next proposition we solve the optimization problems for $F_{1,q}$ in the class of general domains and in that of convex domains.

Proposition 5.3

For $0<q<1$, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \sup \big \{F_{1,q}(\Omega )\ :\ \Omega \subset {\mathbb {R}}^d\text { open and convex },\ 0<|\Omega |<\infty \big \}=+\infty ;\\ \min \big \{F_{1,q}(\Omega )\ :\ \Omega \subset {\mathbb {R}}^d\text { open},\ 0<|\Omega |<\infty \big \}=F_{1,q}(B). \end{array}\right. } \end{aligned}$$

For $q>1$, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \inf \big \{F_{1,q}(\Omega )\ :\ \Omega \subset {\mathbb {R}}^d\text { open and convex},\ 0<|\Omega |<\infty \Big \}=0;\\ \max \big \{F_{1,q}(\Omega )\ :\ \Omega \subset {\mathbb {R}}^d\text { open },\ 0<|\Omega |<\infty \big \}=F_{1,q}(B). \end{array}\right. } \end{aligned}$$

Proof

The minimality (respectively maximality) of B, for $0<q<1$ (respectively for $q>1$), is an immediate consequence of the well known inequality

$$\begin{aligned} h(B)|B|^{1/d}\le h(\Omega )|\Omega |^{1/d}, \end{aligned}$$

which holds for any $\Omega \subset {\mathbb {R}}^d$ with finite measure. To prove the other cases we use the inequality

$$\begin{aligned} h(\Omega )\ge \frac{P(\Omega )}{d |\Omega |}, \end{aligned}$$

which holds for any $\Omega \subset {\mathbb {R}}^d$ open, bounded, convex set (see [8, Corollary 5.2]). Then taking $C_{A,{\varepsilon }}$ as in (3.1) we get

$$\begin{aligned} \lim _{{\varepsilon }\rightarrow 0}h(C_{A,{\varepsilon }})|C_{A,{\varepsilon }}|^{1/d}=+\infty , \end{aligned}$$

from which the thesis easily follows. $\square$

6 The case $p=\infty$

The limit behaviour of the quantities $\lambda _p(\Omega )$, $T_p(\Omega )$, as $p\rightarrow \infty$, are well known for bounded open sets $\Omega \subset {\mathbb {R}}^d$: in [14] and in [21] the authors prove that

$$\begin{aligned} (\lambda _p(\Omega ))^{1/p}\rightarrow \frac{1}{\rho (\Omega )}, \end{aligned}$$

(6.1)

while, following [3] (see also [23]) it holds $w_p\rightarrow d_{\Omega }$ uniformly in $\Omega$, which implies

$$\begin{aligned} (T_p(\Omega ))^{1/p}\rightarrow \int _\Omega d_\Omega (x)\,dx. \end{aligned}$$

(6.2)

Actually, in all these results, the boundedness assumption on $\Omega$ is not needed, as it is only used to provide the compactness of the embedding $W_0^{1,p}(\Omega )$ into the space $C_0(\Omega )$ defined as the completion of $C_c(\Omega )$ with respect to the uniform convergence. Indeed, this holds under the weaker assumption that $|\Omega |<+\infty$ (see Appendix A for more details and for a $\Gamma$-convergence point of view of both limits (6.1) and (6.2)).

According to (6.1) and to (6.2) we define the shape functional $F_{\infty ,q}$ as

(6.3)

Proposition 6.1

Let $\Omega \subset {\mathbb {R}}^d$ be an open convex set. Then

(6.4)

Moreover, both inequalities are sharp. In particular

$$\begin{aligned} {\left\{ \begin{array}{ll} \sup \big \{F_{\infty ,1}(\Omega )\ :\ \Omega \text { open and convex in }{\mathbb {R}}^d,\ 0<|\Omega |<\infty \big \}=1/2;\\ \min \big \{F_{\infty ,1}(\Omega )\ :\ \Omega \text { open and convex in }{\mathbb {R}}^d,\ 0<|\Omega |<\infty \Big \}=F_{\infty ,1}(B)=1/(d+1). \end{array}\right. } \end{aligned}$$

For its proof, we recall the following result, for which we refer to [4, 15].

Theorem 6.2

Let $1\le q\le p$. Then for every convex set E of ${\mathbb {R}}^N$ $(N\ge 1)$ and every nonnegative concave function f on E we have

where the constant $C_{p,q}$ is given by

$$\begin{aligned} C_{p,q}=\left( {\begin{array}{c}N+q\\ N\end{array}}\right) ^{1/q}\left( {\begin{array}{c}N+p\\ N\end{array}}\right) ^{-1/p}. \end{aligned}$$

In addition, the inequality above becomes an equality when E is a ball of radius 1 and $f(x)=1-|x|$.

Proof of Proposition 6.1

In order to prove the right-hand side inequality in (6.4), for every $t\ge 0$, we denote by $\Omega (t)$ the interior parallel set at distance t from $\partial \Omega$, i.e.

$$\begin{aligned} \Omega (t):=\big \{x\in \Omega \ :\ d(x,\partial \Omega )>t\big \}, \end{aligned}$$

and by $A(t):=|\Omega (t)|$. Moreover we set

$$\begin{aligned} L(t):=P(\{x\in \Omega \ :\ d(x,\partial \Omega )=t\}). \end{aligned}$$

Then for a.e. $t\in (0,\rho (\Omega ))$ there exists the derivative $A'(t)$ and it coincides with $-L(t)$. Moreover, being $\Omega$ a convex set, L is a monotone decreasing function. Then A is a convex function such that and $A(0)=|\Omega |$. As a consequence we have

Integrating by parts, we get

The value 1/2 is asymptotically attained in (6.4) by considering a sequence of slab domains

$$\begin{aligned} \Omega _{\varepsilon }:=(0,1)^{d-1}\times (0,{\varepsilon })\subset {\mathbb {R}}^d, \end{aligned}$$

as ${\varepsilon }\rightarrow 0$. Indeed, we have $\rho (\Omega _{{\varepsilon }})={\varepsilon }/2$ and $|\Omega _{{\varepsilon }}|={\varepsilon }$. Being

$$\begin{aligned} A_{{\varepsilon }}(t)=|(t,1-t)^{d-1}\times (t,{\varepsilon }-t)|= (1-2t)^{d-1}({\varepsilon }-2t) \end{aligned}$$

we get

Now we prove the left-hand side inequality in (6.4). Since $\Omega$ is convex, the distance function $d_\Omega$ is concave (see [2]); then, applying Theorem (6.2) to $d_\Omega$, we obtain

(6.5)

Since (6.5) is an identity when $\Omega =B$, $C_{p,1}$ satisfies

$$\begin{aligned} C_{p,1}=\frac{\Vert f\Vert _p}{\Vert f\Vert _1}\omega _d^{1-1/p}\rightarrow \frac{\omega _d}{\Vert f\Vert _1}=(d+1). \end{aligned}$$

As $p\rightarrow \infty$ in (6.5), we obtain

which is an equality when $\Omega =B$. $\square$

Remark 6.3

The proof of the right-hand side of (6.4) relies on the convexity properties of the function A(t). In the planar case a general result, due to Sz. Nagy (see [29]), ensures that, if $\Omega$ is any bounded k-connected open set, (i.e. $\Omega ^c$ has k bounded connected components), then the function

is convex. Therefore, for such an $\Omega$, with the same argument as above it is easy to prove that

Hence, it is interesting to notice how, even when $k=0,1$, the upper bound given in (6.4) remains sharp. In other words, in the maximization of $F_{\infty ,1}$ on planar domains, there is no gain in replacing the class of convex domains by the larger one consisting of simply-connected domains or even more in allowing $\Omega$ to have a single hole.

In the general case $q\ne 1$ the optimization problems for the functional $F_{\infty ,q}$ defined in (6.3) are studied below.

Corollary 6.4

If $0<q<1$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} \sup \big \{F_{\infty ,q}(\Omega )\ :\ \Omega \text { open and convex in }{\mathbb {R}}^d,\ 0<|\Omega |<\infty \big \}=\infty ;\\ \min \big \{F_{\infty ,q}(\Omega )\ :\ \Omega \text { open and convex in }{\mathbb {R}}^d,\ 0<|\Omega |<\infty \Big \}=F_{\infty ,q}(B)=(d+1)^{-q}\omega _d^{(1-q)/d}. \end{array}\right. } \end{aligned}$$

If $q>1$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} \sup \big \{F_{\infty ,q}(\Omega )\ :\ \Omega \text { open and convex in }{\mathbb {R}}^d,\ 0<|\Omega |<\infty \big \}\le (1/2)^q\omega _d ^{(1-q)/d} ;\\ \inf \big \{F_{\infty ,q}(\Omega )\ :\ \Omega \text { open and convex in }{\mathbb {R}}^d,\ 0<|\Omega |<\infty \Big \}=0\;. \end{array}\right. } \end{aligned}$$

Proof

Notice that

$$\begin{aligned} F_{\infty , q}(\Omega )=F_{\infty ,1}^{q}(\Omega )\left( \frac{\rho (\Omega )}{|\Omega |^{1/d}}\right) ^{q-1}, \end{aligned}$$

(6.6)

and that the inequality $|\Omega |\ge \omega _d\rho (\Omega )^d$ holds for every open set $\Omega \subset {\mathbb {R}}^d$ with equality when $\Omega =B$. Thus, if $0<q<1$, by (6.4) we have

$$\begin{aligned} F_{\infty ,q}(\Omega )\ge F_{\infty ,q}(B)=(d+1)^{-q}\omega _d^{(1-q)/d}, \end{aligned}$$

while if $q>1$, using again (6.4) we have

$$\begin{aligned} F_{\infty ,q}(\Omega )\le (1/2)^q\omega _d^{(1-q)/d}. \end{aligned}$$

Finally, let $\Omega _{\varepsilon }$ be the slab domain as in Proposition 6.1. Then

$$\begin{aligned} \lim _{{\varepsilon }\rightarrow 0}F_{\infty ,q}(\Omega _{\varepsilon })=(1/2)^{-q}\lim _{{\varepsilon }\rightarrow 0}\left( \frac{{\varepsilon }}{2{\varepsilon }^{1/d}}\right) ^{q-1}= {\left\{ \begin{array}{ll} 0,&{}\hbox {if}\quad q>1;\\ \infty ,&{}\hbox {if}\quad 0<q<1, \end{array}\right. } \end{aligned}$$

from which the thesis is achieved. $\square$

If we remove the convexity assumption on the admissible domains $\Omega$, the picture is similar to those provided by Proposition 2.1 and Proposition 2.3. More precisely, if $q>1/(d+1)$, the minimization problem for $F_{\infty ,q}$ is ill posed. When $q>1$, this follows directly by Corollary 6.4, while, in the case $1/(d+1)<q\le 1$, by taking $\Omega _n$ to be the union of n disjoint balls of radius $r_j=j^{-1/d}$ with $j=1,\dots ,n$, one can verify that $F_{\infty ,q}(\Omega _n)\rightarrow 0$, as $n\rightarrow \infty$. On the contrary, when $q\le 1/(d+1)$, the minimum of $F_{\infty ,q}$ is attained by any ball. Indeed, since $1/(d+1)\le p'/(p'+d)$ for every $p>1$, by using Proposition 2.1, we have

$$\begin{aligned} F^{1/p}_{p,q}(\Omega )\ge F^{1/p}_{p,q}(B). \end{aligned}$$

Hence, passing to the limit as $p\rightarrow +\infty$, we obtain

$$\begin{aligned} F_{\infty ,q}(\Omega )\ge F_{\infty ,q}(B). \end{aligned}$$

Concerning the upper bound, Corollary 6.4 implies that the maximization problem is ill posed in the case $q<1$, while, when $q\ge 1$, using (6.6), we obtain

$$\begin{aligned} F_{\infty ,q}(\Omega )\le \omega _d^{(1-q)/d}. \end{aligned}$$

However, working with general domains provides an upper bound larger than in (6.4); for instance, in the two-dimensional case, taking as $\Omega _N$ the unit disk where we remove N points as in Fig. 1, gives

where E is the regular exagon with unitary sides centered at the origin, as an easy calculation shows.

Fig. 1

The two-dimensional region $\Omega _N$

7 Further remarks and open questions

Several interesting problems and questions about the shape functionals $F_{p,q}$ are still open; in this section we list some of them.

Problem 1

The characterization of the infimum of $F_{p,q}$ in the class of all domains is well clarified in Proposition 2.1; on the contrary, for the supremum of $F_{p,q}$, Proposition 2.3 only says it is finite for $q\ge 1$. It would be interesting to know if the supremum can be better characterized, if it is attained, and in particular if it is attained for a ball when the exponent q is large enough (see also Problems 1 and 2 in [30]).

Problem 2

Concerning Problem 1 above, the case $q=1$ is particularly interesting. The Polya inequality gives $\sup F_{p,1}\le 1$, and Proposition 2.4 gives $\sup F_{p,1}=1$ whenever $p\le d$. It would be interesting to prove (or disprove) that $\sup F_{p,1}<1$ for all $p>d$.

Problem 3

In the convex setting, Proposition 3.2 provides some upper and lower bounds to $F_{p,1}$ that however are far from being sharp. Even in the case $p=2$, sharp values for the infimum and the supremum of $F_{2,1}$ in the class of convex sets are unknown (see Conjecture 4.2. in [30]). It seems natural to conjecture that the right sharp inequalities are those given in Theorem 4.4 for $F_{p,1}$ on the class of thin domain.

Problem 4

In the two-dimensional case with $p=\infty$ we have seen that the domains $\Omega _N$ in Fig. 1 give the asymptotic value $\frac{1}{3}+\frac{\log 3}{4}$ for the shape functional $F_{\infty ,1}$. It would be interesting to prove (or disprove) that this number is actually the supremum of $F_{\infty ,1}(\Omega )$ when $\Omega$ varies in the class of all bounded open two-dimensional sets. In addition, in the case of a dimension $d>2$, it is not clear how a maximizing sequence $(\Omega _n)$ for $F_{\infty ,1}$ has to be.

Appendix A

We devote this Appendix to give a proof of the known asymptotics (6.1) and (6.2) by means of $\Gamma$-convergence when $0<|\Omega |<+\infty$. We recall that if $p>d$ and $\Omega \subset {\mathbb {R}}^d$ is any (possibly unbounded) open set with finite measure we have the compact embedding:

$$\begin{aligned} W^{1,p}_0(\Omega )\hookrightarrow C_0(\Omega ). \end{aligned}$$

(A.1)

A quick proof of it can be obtained by combining the Gagliardo–Niremberg inequality in $W^{1,p}({\mathbb {R}}^d)$:

$$\begin{aligned} \Vert \phi \Vert _{L^{\infty }({\mathbb {R}}^d)}\le C(d,p)\Vert \nabla \phi \Vert ^{1-\alpha }_{L^{p}({\mathbb {R}}^d)}\Vert \phi \Vert ^{\alpha }_{L^{p}({\mathbb {R}}^d)}\,\quad \forall \phi \in W^{1,p}({\mathbb {R}}^d), \end{aligned}$$

(A.2)

together with the well known facts that the inclusion $W^{1,p}_0(\Omega )\hookrightarrow L^p(\Omega )$ is compact (thanks to the Riesz–Frechét–Kolmogorov Theorem) and the embedding (A.1) is continuous. Note that, in (A.2), C(d, p) denotes a positive constant depending on p and d, and $\alpha =1-d/p$ (we refer to [1], Chapter 6, for a comprehensive discussion on necessary and sufficient conditions for the compactness of several embeddings of Sobolev spaces).

In particular, if we denote by $W_0^{1,\infty }(\Omega )$ the closure of $C_c^\infty (\Omega )$ with respect to the weak* convergence of $W^{1,\infty }(\Omega )$, we have that $u\in W^{1,\infty }_0(\Omega )$ if and only if $u\in C_0(\Omega )$ and u is a Lipschitz continuous function on $\Omega$. Moreover $W_0^{1,\infty }(\Omega )$ can be easily characterized as:

$$\begin{aligned} W^{1,\infty }_0(\Omega )=W^{1,\infty }(\Omega )\cap \bigcap _{p\ge 1}W^{1,p}_0(\Omega ). \end{aligned}$$

(A.3)

Proposition A.1

Let $\Omega \subset {\mathbb {R}}^d$ be an open set with finite measure and let $\Psi _p,\Psi _\infty : L^{1}(\Omega )\rightarrow {{\bar{{\mathbb {R}}}}}$ be defined by

$$\begin{aligned} \begin{aligned}&\Psi _p(u):={\left\{ \begin{array}{ll} \Vert \nabla u\Vert _{L^{p}(\Omega )}&{}\hbox {if}\quad u\in W_0^{1,p}(\Omega )\quad \text {and}\quad \Vert u\Vert _p=1,\\ +\infty &{}\hbox {otherwise}, \end{array}\right. }\\&\Psi _\infty (u):= {\left\{ \begin{array}{ll} \Vert \nabla u\Vert _{L^{\infty }(\Omega )}&{}\hbox {if}\quad u\in W^{1,\infty }_0(\Omega ),\ \Vert u\Vert _\infty =1,\\ +\infty &{}\hbox {otherwise}. \end{array}\right. } \end{aligned} \end{aligned}$$

Then, as $p\rightarrow \infty$, the sequence $\Psi _p$ $\Gamma$-converges to $\Psi _\infty$ with respect to the $L^1$-convergence.

Proof

Let $p_n\rightarrow \infty$. The $\Gamma$-$\limsup$ inequality is trivial since, for every $u\in W^{1,\infty }_0(\Omega )$ with $\Vert u\Vert _\infty =1$, the sequence $u_{p_n}=\Vert u\Vert _{p_n}^{-1}u$ converges to u in $L^1$ and satisfies

$$\begin{aligned} \Vert u_n\Vert _{p_n}=1,\quad \limsup _{n\rightarrow \infty }\Psi _{p_n} (u_n)= \Psi _\infty (u). \end{aligned}$$

To prove the $\Gamma$-$\liminf$ inequality, without loss of generality, let $u\in L^{\infty }(\Omega )$, $(u_{p_n})\subseteq W^{1,p_n}_0(\Omega )$ be such that $u_{p_n}\rightarrow u$ in $L^1(\Omega )$, $\Vert u_{p_n}\Vert _{p_n}=1$, and $\liminf _{n\rightarrow \infty }\Psi _{p_n}(u_{p_n})=C<\infty$. Since for every $q\ge 1$ and for n large enough it holds

$$\begin{aligned} \Vert Du_{p_n}\Vert _{q}\le |\Omega |^{ 1/q-1/{p_n}}\Psi _{p_n}(u_{p_n}) \end{aligned}$$

we get that $u_{p_n}\rightarrow u$ in $L^q(\Omega )$, $u\in W^{1,q}_0(\Omega )$ and

$$\begin{aligned} \Vert Du\Vert _{q}\le C |\Omega |^{ 1/q}. \end{aligned}$$

(A.4)

Moreover

$$\begin{aligned} |\Omega |^{1/q}=\lim _{n\rightarrow \infty }|\Omega |^{1/q-1/p_n}\Vert u_{p_n}\Vert _p\ge \lim _{n\rightarrow \infty }\Vert u_{p_n}\Vert _q=\Vert u\Vert _q, \end{aligned}$$

which yields, as $q\rightarrow \infty$, $\Vert u\Vert _{\infty }\le 1$. Combining this estimate with (A.4), we get that $u\in W^{1,\infty }(\Omega )$; hence, by (A.3), $u\in W^{1,\infty }_0(\Omega )$. Thanks to the compact embedding of $W_0^{1,q}(\Omega )$ in $C_0(\Omega )$ when $q>d$, we obtain that $\Vert u_{p_n}- u\Vert _{\infty }\rightarrow 0$ as $n\rightarrow \infty$ and, since

$$\begin{aligned} 1=\Vert u_n\Vert _{p_n}\le |\Omega |^{1/p_n}\Vert u_{p_n}\Vert _\infty , \end{aligned}$$

we get that $\Vert u\Vert _\infty =1$. Finally, by letting $n\rightarrow \infty$ in (A.4), it follows

$$\begin{aligned} \Psi _\infty (u)=\Vert Du\Vert _\infty \le C=\liminf _{n\rightarrow \infty }\Psi (u_{p_n}) . \end{aligned}$$

$\square$

Corollary A.2

Let $\Omega \subset {\mathbb {R}}^d$ be an open set with finite measure. Then, as $p\rightarrow \infty$,

$$\begin{aligned} \lambda ^{1/p}_p(\Omega )\rightarrow \rho (\Omega )^{-1}. \end{aligned}$$

Proof

Using $d_{\Omega }$ as a test function for (1.6) we have $\limsup _{p\rightarrow \infty }\lambda _p(\Omega )\le \rho (\Omega )^{-1}$. Moreover we notice that, for any $\phi \in W^{1,\infty }_0(\Omega )$, it holds

$$\begin{aligned} |\phi (x)|\le d_{\Omega }(x)\Vert \nabla \phi \Vert _{\infty }. \end{aligned}$$

(A.5)

Let $u_p$ be the (only) minimum of $\Psi _p$, and let $p_n\rightarrow \infty$. With the same argument of Proposition A.1 we can assume $u_{p_n}\rightarrow u_\infty$ uniformly in $L^{1}(\Omega )$, where $u_\infty \in W^{1,\infty }_0(\Omega )$ and $\Vert u_{\infty }\Vert _{\infty }=1$. Then, by Proposition A.1 and by (A.5), we have

$$\begin{aligned} \rho (\Omega )^{-1}\le \Vert \nabla u_{\infty }\Vert =\Psi (u_\infty )\le \liminf _{n\rightarrow \infty }\Psi (u_{p_n})=\liminf _{p\rightarrow \infty }\lambda _{p_n}^{1/p_n}(\Omega ). \end{aligned}$$

The thesis follows by the arbitrariness of the sequence $p_n$. $\square$

Next Proposition generalizes Proposition 2.1 in [16].

Proposition A.3

Let $\Omega \subset {\mathbb {R}}^d$ be an open set with finite measure and let $\Phi _p,\Phi _\infty :L^1(\Omega )\rightarrow {{\bar{{\mathbb {R}}}}}$ be defined by

$$\begin{aligned} \begin{aligned}&\Phi _p(u):={\left\{ \begin{array}{ll} \frac{1}{p}\int _\Omega |\nabla u|^p\,dx&{}\hbox {if}\quad u\in W_0^{1,p}(\Omega ),\\ +\infty &{}\hbox {otherwise}, \end{array}\right. }\\&\Phi _\infty (u):={\left\{ \begin{array}{ll} 0&{}\hbox {if}\quad u\in W^{1,\infty }_0(\Omega ),\ \Vert \nabla u\Vert _\infty \le 1,\\ +\infty &{}\hbox {otherwise}. \end{array}\right. } \end{aligned} \end{aligned}$$

Then, as $p\rightarrow \infty$, the functionals $\Phi _p$ $\Gamma$-converge to $\Phi _\infty$ with respect to the $L^1$-convergence. Moreover, we have

$$\begin{aligned} \lim _{p\rightarrow \infty }\Phi _p(u)=\Phi _\infty (u)\quad \hbox {for every}\quad u\in L^1(\Omega ). \end{aligned}$$

Proof

Let $p_n\rightarrow \infty$. The $\Gamma$-limsup inequality is trivial since for any $u\in W^{1,\infty }_0(\Omega )$ with $\Vert \nabla u\Vert _\infty \le 1$ we have $u\in W_0^{1,p_n}(\Omega )$ for every $n\in {\mathbb {N}}$ and thus

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Phi _{p_n}(u)=\limsup _{n\rightarrow \infty }\int _\Omega \frac{1}{p_n}|\nabla u(x)|^{p_n}\,dx\le |\Omega |\limsup _{n\rightarrow \infty }\frac{1}{p_n}=0=\Phi _\infty (u). \end{aligned}$$

To prove the $\Gamma$-liminf inequality, we can assume $u_n,u\in L^1(\Omega )$, $u_n\rightarrow u$ in $L^1(\Omega )$, $u_n\in W_0^{1,p_n}(\Omega )$, and

$$\begin{aligned} \liminf _{n\rightarrow \infty }\Phi _{p_n}(u_{p_n})=\lim _{n\rightarrow \infty }\Phi _{p_n}(u_{p_n})=M<+\infty . \end{aligned}$$

If $q>1$ and $p_n>q$ we have

$$\begin{aligned} \Vert \nabla u_{p_n}\Vert _q\le |\Omega |^{1/q-1/{p_n}}\Vert \nabla u_{p_n}\Vert _{p_n}\le |\Omega |^{1/q-1/{p_n}}(M p_n)^{1/p_n} \end{aligned}$$

which forces $u\in W^{1,q}_0(\Omega )$. Moreover, since

$$\begin{aligned} \Vert \nabla u\Vert _q\le \liminf _{n\rightarrow \infty }\Vert \nabla u_{p_n}\Vert _q\le |\Omega |^{1/q}, \end{aligned}$$

we have also $\Vert \nabla u\Vert _\infty \le 1$. Therefore, by (A.3), $u\in W_0^{1,\infty }(\Omega )$ and

$$\begin{aligned} \Phi _\infty (u)=0\le \liminf _{n\rightarrow \infty }\Phi _{p_n}(u_{p_n}). \end{aligned}$$

The thesis follows by the arbitrariness of the sequence $p_n$. $\square$

Corollary A.4

Let $\Omega \subset {\mathbb {R}}^d$ be an open set with finite measure, let $w_p$ be the solution to (1.2). Then, as $p\rightarrow \infty$,

$$\begin{aligned} w_p\rightarrow d_\Omega \ \text {in } L^\infty (\Omega ), \quad (T_p(\Omega ))^{1/p}\rightarrow \int _\Omega d_\Omega (x)\,dx. \end{aligned}$$

Proof

It is sufficient to show that $w_p\rightarrow d_\Omega$ uniformly in $\Omega$. First we notice that by (2.2) we get

$$\begin{aligned} \Big (|\Omega |^{-1}\int _\Omega |\nabla w_{p}|^{p}\,dx\Big )^{1/p}=\big (|\Omega |^{1-p}T_{p_n}(\Omega )\big )^{1/(p(p-1))}\le \Big (\lambda ^{1/p}_{p}(\Omega )\Big )^{1/1-p}. \end{aligned}$$

(A.6)

By Corollary A.2 we have

$$\begin{aligned} C:=\sup _p|\Omega |^{-1/{p}}\Vert \nabla w_{p}\Vert _{p}<+\infty . \end{aligned}$$

Moreover, for every fixed $q\ge 1$ and p large enough, by Hölder inequality, we have that

$$\begin{aligned} \Vert \nabla w_{p}\Vert _q\le \Vert \nabla w_{p}\Vert _{p}|\Omega |^{1/q-1/{p}}\le C|\Omega |^{1/q}. \end{aligned}$$

(A.7)

Let $p_n\rightarrow \infty$. By applying (A.7) we can show that there exists $w_{\infty }\in W^{1,\infty }_0(\Omega )$, such that $w_{p_n}$ converges uniformly to $w_\infty$ and weakly in $W^{1,q}(\Omega )$ for every $q\ge 1$. Notice that (A.6) combined with Corollary A.2 shows also

$$\begin{aligned} \Vert \nabla w_\infty \Vert _q\le |\Omega |^{1/q}, \end{aligned}$$

for every $q\ge 1$, which implies $\Vert w_\infty \Vert \le 1$.

Now let $J_p$ be the functional defined in (1.4) and $J_\infty$ be the functional given by

$$\begin{aligned} J_\infty (u):=\Phi _{\infty }(u)-\int _\Omega u\,dx. \end{aligned}$$

Since the functional $u\mapsto \int _\Omega u\,dx$ is continuous with respect to the $L^1$-convergence, thanks to Proposition A.3, we have that

$$\begin{aligned} \lim _{n\rightarrow \infty }J_{p_n}(w_{p_n})=J_\infty (w_\infty )=\min _{u\in W_0^{1,\infty }(\Omega ), \Vert \nabla u\Vert _\infty \le 1}J_\infty (u)=-\int _\Omega w_\infty \,dx. \end{aligned}$$

Moreover, by using (A.5) we have $w_\infty (x)\le d_\Omega (x)$. In addition, since $d_\Omega \in W^{1,\infty }_0(\Omega )$, we have also $J_\infty (w_\infty )\le J_\infty (d_\Omega )$, i.e. $\int _\Omega (w_\infty -d_\Omega )\,dx\ge 0$. Hence $d_\Omega =w_\infty$. By the arbitrariness of the sequence $p_n$, we get that $w_p\rightarrow d_{\Omega }$ uniformly as $p\rightarrow \infty$. $\square$