Torsional rigidity in random walk spaces

TL;DR

This paper investigates nonlocal torsional rigidity in random walk spaces, establishing its relation with spectral heat content, eigenvalues, and inequalities, thus extending classical results to nonlocal and graph settings.

Contribution

It introduces a comprehensive framework connecting nonlocal torsional rigidity with spectral properties, inequalities, and graph structures, providing new insights and generalizations.

Findings

Relation between nonlocal torsional rigidity and spectral m-heat content

Recovery of the first eigenvalue via a limit formula

Nonlocal Saint-Venant and Pólya-Makai inequalities

Abstract

In this paper we study the (nonlocal) torsional rigidity in the ambient space of random walk spaces. We get the relation of the (nonlocal) torsional rigidity of a set $\Omega$ with the spectral $m$-heat content of $\Omega$, what gives rise to a complete description of the nonlocal torsional rigidity of $\Omega$ by using uniquely probability terms involving the set $\Omega$; and recover the first eigenvalue of the nonlocal Laplacian with homogeneous Dirichlet boundary conditions by a limit formula using these probability term. For the random walk in $\R^N$ associated with a non singular kernel, we get a nonlocal version of the Saint-Venant inequality, and, under rescaling we recover the classical Saint-Venant inequality. We study the nonlocal $p$-torsional rigidity and its relation with the nonlocal Cheeger constants. We also get a nonlocal version of the P\'{o}lya-Makai-type inequalities. We relate the torsional rigidity given here for weighted graphs with the torsional rigidity on metric graphs.

Equations

$$\left\{\begin{array}[]{ll}-\Delta u_{D}=1\quad\hbox{in}\ D\\[10.0pt] u_{D}=0\quad\hbox{on}\ \partial D.\end{array}\right.$$

$$E(D)=\underset{v\in W_0^{1,2}(D)}{\min }\frac{1}{2}{\int }_D|\nabla v{|}^{2}dx-{\int }_{D}vdx.$$

$$T(D):={\int }_{\Omega }{u}_D(x)dx$$

$$T(D)=\underset{v\in W_0^{1,2}(D)\setminus \{0\}}{\max }\frac{{\left({\int }_{D}vdx\right)}^2}{{\int }_D|\nabla v{|}^{2}dx}.$$

$$T(D)\le T(D^{\ast }).$$

$${\lambda }_1(D^{\ast })\le {\lambda }_1(D),$$

$$\left\{\begin{array}[]{ll}-\Delta u=\lambda u,\quad\hbox{in}\ D\\[10.0pt] u=0,\quad\hbox{on}\ \partial D.\end{array}\right.$$

$${\lambda }_1(D)=\underset{v\in W_0^{1,2}(D)\setminus \{0\}}{\min }\frac{{\int }_D|\nabla v{|}^{2}dx}{{\int }_D{v}^{2}dx},$$

$${\lambda }_1(D)\le \frac{|D|}{T(D)}.$$

$${\mathbb{Q}}_D(t):={\int }_D{v}_D(x,t)dx$$

$$\left\{\begin{array}[]{lll}\displaystyle\frac{v_{D}(x,t)}{\partial t}(x,t)=\Delta v_{D}(x,t)(x,t),&\hbox{if}\ (x,t)\in D\times[0,\infty),\\[10.0pt] v_{D}(x,t)(x,t)=0,&\hbox{if}\ (x,t)\in\partial D\times(0,\infty),\\[10.0pt] v_{D}(x,t)(x,0)=\raisebox{2.0pt}{\rm{$\chi$}}_{D}(x),&\hbox{if}\ x\in D.\end{array}\right.$$

$$\tau =\inf \{s\ge 0:B(s)\notin D\}.$$

$$u_D(x)={\mathbb{E}}_x[\tau ],x\in D,$$

$$v_D(x,t)={\mathbb{P}}_x[\tau >t],x\in D,t>0.$$

$$E{M}_j(D):={\int }_D{\mathbb{E}}_x[{\tau }^j]dx.$$

$$T(D)=E{M}_1(D).$$

$${\mathbb{E}}_x[{\tau }^j]=j{\int }_0^{\infty }{t}^{j-1}{v}_D(x,t)dt.$$

$$E{M}_j(D)=j{\int }_0^{\infty }{t}^{j-1}{\mathbb{Q}}_D(t)dt.$$

$$T(D)={\int }_0^{\infty }{\mathbb{Q}}_D(t)dt.$$

$$d{m}_x^{\ast n}(y):={\int }_{z\in X}d{m}_z(y)d{m}_x^{\ast (n-1)}(z),n\ge 1$$

$$\mu \ast m(A):={\int }_X{m}_x(A)d\mu (x)\forall A\in \mathcal{B},$$

$$\nu \ast m=\nu .$$

$$d{m}_x(y)d\nu (x)=d{m}_y(x)d\nu (y)$$

$$\sum _{n=1}^{\infty }{m}_x^{\ast n}(D)>0.$$

$$L_m(A,B):={\int }_A{\int }_{B}d{m}_x(y)d\nu (x)={\int }_A{m}_x(B)d\nu (x).$$

$${\mathcal{B}}_{\Omega }:=\{B\in \mathcal{B}:B\subset \Omega \}.$$

$$m_x^J(A):={\int }_{A}J(x-y)dyfor\; everyx\in {\mathbb{R}}^{N}and\; every\; Borel\; setA\subset {\mathbb{R}}^N.$$

$$G=(V(G),E(G)),$$

$$d_x:=\sum _{y\sim x}{w}_{xy}=\sum _{y\in V(G)}{w}_{xy},$$

$$m_x^G:=\frac{1}{d_x}\sum _{y\sim x}{w}_{xy}{\delta }_y.$$

Full text

2010 Mathematics Subject Classification: 35K55, 47H06, 47J35.

1. Introduction

In this paper we study the (nonlocal) torsional rigidity in the ambient space of random walk spaces. Important examples of these spaces are locally finite weighted graphs, finite Markov chains and nonlocal operators on domains in $\mathbb{R}^{N}$ where the jumps are driven by a non-negative integrable and radially symmetric kernel (see [30] and [32]).

In the classical local setting, the torsional rigidity of a Lebesgue subset of $\mathbb{R}^{N}$ has been, and is nowadays, a source of interesting problems. Let us consider an isotropic elastic cylindrical beam in $\mathbb{R}^{3}$ with cross-section, perpendicular to the $z$-axis, is an an open bounded domain $D\subset\mathbb{R}^{2}$. The torsion rigidity problem (see e.g. [44]) is to find the shape of the cross section $D$ which provides the greatest torsional rigidity, under an area constraint, when a torque is applied around the $z$-axis . It was conjectured by A. Saint-Venant in 1856 that the simply connected cross-section with maximal torsional rigidity is the circle and it was proved by G. Pólya in 1948. The distribution of stress generated in the beam due to the applied torque is determined by the stress function $u_{D}$, the unique positive weak solution of the Dirichlet problem

[TABLE]

Notice that the function $u_{D}$ is also the unique minimizer of the torsional energy

[TABLE]

The total resultant torque due to this stress function is called torsional rigidity and is expressed as

[TABLE]

or equivalently (see [37] or [3])

[TABLE]

Throughout this paper, we adopt the following notation. If $D$ is open in $\mathbb{R}^{N}$ with $0<|D|<\infty$ then $D^{*}$ is the ball in $\mathbb{R}^{N}$ centered at the origin with $|D^{*}|=|D|$. Furthermore $B_{R}$ is a ball with radius $R$. We put $\omega_{N}=|B_{1}|$.

The Saint-Venant inequality reads, for $D$ a bounded domain, as follows:

[TABLE]

This inequality was established by G. Pólya [37] using symmetrization methods (see also E. Makai [29]).

On the other and, the Faber-Krahn inequality establishes that

[TABLE]

where $\lambda_{1}(D)$ is the lowest $\lambda$ for which the eigenvalue problem

[TABLE]

admits a non trivial solution. The first proof of the Faber-Krahn inequality was given by Pólya and Szegö in [38] based in spherically symmetric decreasing rearrangement.

Since $\lambda_{1}(D)$ is the minimizer of the Rayleigh quotient

[TABLE]

is easy to see (see, for example, [8]) that

[TABLE]

Let $D$ be an open bounded domain $D\subset\mathbb{R}^{N}$. The spectral heat content of $D$ is given by

[TABLE]

where $v_{D}$ is the solution of Dirichlet problem

[TABLE]

$\mathbb{Q}_{D}(t)$ represents the amount of heat contained in $D$ at time $t$ when $D$ has initial temperature $1$ and when the boundary of $D$ is keps at temperature [math] for all $t>0$.

The functions $u_{D}$ and $v_{D}$ have a probabilistic interpretation (see for instance [6]). For this, let $(B(s),s\geq 0,\mathbb{P}_{x},x\in\mathbb{R}^{N})$ be a brownian motion associated to the Laplacian on $\mathbb{R}^{N}$, and let $\tau$ be the first exit time from $D$:

[TABLE]

Then

[TABLE]

where $\mathbb{E}_{x}$ denotes expectation with respect to $\mathbb{P}_{x}$, and

[TABLE]

For $j\in\mathbb{N}$ the sequence of exit-moments of $D$ is defined as

[TABLE]

Notice that, by (1.5),

[TABLE]

Using (1.6), we can express moments of the exit time in term of $v_{D}$ as

[TABLE]

Integrating in (1.8) and using Fubini’s Theorem, we see that the sequence of exit-moments can be expressed as moments of the heat content:

[TABLE]

In particular, by (1.7), we have

[TABLE]

Our aim is to study the torsional rigidity in the general framework of the random walk spaces. We get the nonlocal versions of the previous local results (1.2), (1.4), (1.7) and (1.10). In particular we give the precise characterization of the nonlocal torsional rigidity of a set, and of the all nonlocal exit moments, by using uniquely probability terms involving the set, see (3.13) and (3.14), and recover the first eigenvalue of the nonlocal Laplacian with homogeneous Dirichlet boundary conditions, when exists, by a limit formula using such terms, see (3.22). For the random walk in $\mathbb{R}^{N}$ associated with a non singular kernel, we get a nonlocal version of the Saint-Venant inequality, and, under rescaling we recover the classical Saint-Venant inequality. We also get the variational characterization of the nonlocal $p$-torsional rigidity. We relate the nonlocal $p$-torsional rigidity of a set with its $1$-Cheeger and $p$-Cheeger constants in (6.14), and as a consequence we prove that the nonlocal $1$-Cheeger constant of a set is the limit, as $p\to 1^{+}$, of the inverse of its nonlocal $p$-torsional rigidities, see (6.15). See also (6.32) for another limit attaining the nonlocal $1$-Cheeger constant by means of nonlocal Poincaré constants. We also obtain a nonlocal version of Pólya-Makai-type inequalities. To the best of our knowledge most of the results we get are new even for the particular cases of locally finite weighted graphs and nonlocal problems in domains of $\mathbb{R}^{N}$. Finally we relate the torsional rigidity given here for graphs with the torsional rigidity on metric graphs stated in [35].

2. Preliminaries

2.1. Random walk spaces

We recall some concepts and results about random walk spaces given in [30], [31] and [32].

Let $(X,\mathcal{B})$ be a measurable space such that the $\sigma$-field $\mathcal{B}$ is countably generated. A random walk $m$ on $(X,\mathcal{B})$ is a family of probability measures $(m_{x})_{x\in X}$ on $\mathcal{B}$ such that $x\mapsto m_{x}(B)$ is a measurable function on $X$ for each fixed $B\in\mathcal{B}$.

The notation and terminology chosen in this definition comes from Ollivier’s paper [36]. As noted in that paper, geometers may think of $m_{x}$ as a replacement for the notion of balls around $x$, while in probabilistic terms we can rather think of these probability measures as defining a Markov chain whose transition probability from $x$ to $y$ in $n$ steps is

[TABLE]

and $m_{x}^{*0}=\delta_{x}$, the dirac measure at $x$.

Definition 2.1**.**

If $m$ is a random walk on $(X,\mathcal{B})$ and $\mu$ is a $\sigma$-finite measure on $X$. The convolution of $\mu$ with $m$ on $X$ is the measure defined as follows:

[TABLE]

which is the image of $\mu$ by the random walk $m$.**

Definition 2.2**.**

If $m$ is a random walk on $(X,\mathcal{B})$, a $\sigma$-finite measure $\nu$ on $X$ is invariant with respect to the random walk $m$ if

[TABLE]

The measure $\nu$ is said to be reversible if moreover, the detailed balance condition

[TABLE]

holds true.**

Definition 2.3**.**

Let $(X,\mathcal{B})$ be a measurable space where the $\sigma$-field $\mathcal{B}$ is countably generated. Let $m$ be a random walk on $(X,\mathcal{B})$ and $\nu$ an invariant measure with respect to $m$. The measurable space together with $m$ and $\nu$ is then called a random walk space and is denoted by $[X,\mathcal{B},m,\nu]$.**

If $(X,d)$ is a Polish metric space (separable completely metrizable topological space), $\mathcal{B}$ is its Borel $\sigma$-algebra and $\nu$ is a Radon measure (i.e. $\nu$ is inner regular and locally finite), then we denote $[X,\mathcal{B},m,\nu]$ as $[X,d,m,\nu]$, and call it a metric random walk space.

Definition 2.4**.**

Let $[X,\mathcal{B},m,\nu]$ be a random walk space. We say that $[X,\mathcal{B},m,\nu]$ is $m$-connected if, for every $D\in\mathcal{B}$ with $\nu(D)>0$ and $\nu$-a.e. $x\in X$,

[TABLE]

Definition 2.5**.**

Let $[X,\mathcal{B},m,\nu]$ be a random walk space and let $A$, $B\in\mathcal{B}$. We define the $m$-interaction between $A$ and $B$ as

[TABLE]

The following result gives a characterization of $m$-connectedness in terms of the $m$-interaction between sets.

Proposition 2.6**.**

([30, Proposition 2.11], [32, Proposition 1.34]) Let $[X,\mathcal{B},m,\nu]$ be a random walk space. The following statements are equivalent:

(i) $[X,\mathcal{B},m,\nu]$ is $m$-connected.

(ii) If $A,B\in\mathcal{B}$ satisfy $A\cup B=X$ and $L_{m}(A,B)=0$, then either $\nu(A)=0$ or $\nu(B)=0$.

(iii) If $A\in\mathcal{B}$ is a $\nu$-invariant set then either $\nu(A)=0$ or $\nu(X\setminus A)=0$.

Definition 2.7**.**

Let $[X,\mathcal{B},m,\nu]$ be a reversible random walk space, and let $\Omega\in\mathcal{B}$ with $\nu(\Omega)>0$. We denote by $\mathcal{B}_{\Omega}$ to the following $\sigma$-algebra

[TABLE]

We say that $\Omega$ is $m$-connected (with respect to $\nu$) if $L_{m}(A,B)>0$ for every pair of non-$\nu$-null sets $A$, $B\in\mathcal{B}_{\Omega}$ such that $A\cup B=\Omega$.

Let us see now some examples of random walk spaces.

Example 2.8**.**

Consider the metric measure space $(\mathbb{R}^{N},d,\mathcal{L}^{N})$, where $d$ is the Euclidean distance and $\mathcal{L}^{N}$ the Lebesgue measure on $\mathbb{R}^{N}$ (which we will also denote by $|.|$). For simplicity, we will write $dx$ instead of $d\mathcal{L}^{N}(x)$. Let $J:\mathbb{R}^{N}\to[0,+\infty[$ be a measurable, nonnegative and radially symmetric function verifying $\int_{\mathbb{R}^{N}}J(x)dx=1$. Let $m^{J}$ be the following random walk on $(\mathbb{R}^{N},d)$:

[TABLE]

Then, applying Fubini’s Theorem it is easy to see that the Lebesgue measure $\mathcal{L}^{N}$ is reversible with respect to $m^{J}$. Therefore, $[\mathbb{R}^{N},d,m^{J},\mathcal{L}^{N}]$ is a reversible metric random walk space.

Example 2.9**.**

*[Weighted discrete graphs] *Consider a locally finite weighted discrete graph

[TABLE]

where $V(G)$ is the vertex set, $E(G)$ is the edge set and each edge $(x,y)\in E(G)$ (we will write $x\sim y$ if $(x,y)\in E(G)$) has a positive weight $w_{xy}=w_{yx}$ assigned. Suppose further that $w_{xy}=0$ if $(x,y)\not\in E(G)$. Note that there may be loops in the graph, that is, we may have $(x,x)\in E(G)$ for some $x\in V(G)$ and, therefore, $w_{xx}>0$. Recall that a graph is locally finite if every vertex is only contained in a finite number of edges.

A finite sequence $\{x_{k}\}_{k=0}^{n}$ of vertices of the graph is called a path if $x_{k}\sim x_{k+1}$ for all $k=0,1,...,n-1$. The length of a path $\{x_{k}\}_{k=0}^{n}$ is defined as the number $n$ of edges in the path. With this terminology, $G=(V(G),E(G))$ is said to be connected if, for any two vertices $x,y\in V$, there is a path connecting $x$ and $y$, that is, a path $\{x_{k}\}_{k=0}^{n}$ such that $x_{0}=x$ and $x_{n}=y$. Finally, if $G=(V(G),E(G))$ is connected, the graph distance $d_{G}(x,y)$ between any two distinct vertices $x,y$ is defined as the minimum of the lengths of the paths connecting $x$ and $y$. Note that this metric is independent of the weights.

For $x\in V(G)$ we define the weight at $x$ as

[TABLE]

and the neighbourhood of $x$ as $N_{G}(x):=\{y\in V(G)\,:\,x\sim y\}$. Note that, by definition of locally finite graph, the sets $N_{G}(x)$ are finite. When all the weights are $1$, $d_{x}$ coincides with the degree of the vertex $x$ in a graph, that is, the number of edges containing $x$.

For each $x\in V(G)$ we define the following probability measure

[TABLE]

It is not difficult to see that the measure $\nu_{G}$ defined as

[TABLE]

is a reversible measure with respect to this random walk. Therefore, $[V(G),\mathcal{B},m^{G},\nu_{G}]$ is a reversible random walk space being $\mathcal{B}$ is the $\sigma$-algebra of all subsets of $V(G)$. Moreover $[V(G),d_{G},m^{G},\nu_{G}]$ is a reversible metric random walk space.

Example 2.10**.**

Given a random walk space $[X,\mathcal{B},m,\nu]$ and $\Omega\in\mathcal{B}$ with $\nu(\Omega)>0$, let

[TABLE]

Then, $m^{\Omega}$ is a random walk on $(\Omega,\mathcal{B}_{\Omega})$ and it easy to see that $\nu\!\!\mathop{\hbox{ \vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega$ is invariant with respect to $m^{\Omega}$. Therefore, $[\Omega,\mathcal{B}_{\Omega},m^{\Omega},\nu\!\!\mathop{\hbox{ \vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega]$ is a random walk space. Moreover, if $\nu$ is reversible with respect to $m$ then $\nu\!\!\mathop{\hbox{ \vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega$ is reversible with respect to $m^{\Omega}$. Of course, if $\nu$ is a probability measure we may normalize $\nu\!\!\mathop{\hbox{ \vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega$ to obtain the random walk space

[TABLE]

Note that, if $[X,d,m,\nu]$ is a metric random walk space and $\Omega$ is closed, then $[\Omega,d,m^{\Omega},\nu\!\!\mathop{\hbox{ \vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega]$ is also a metric random walk space, where we abuse notation and denote by $d$ the restriction of $d$ to $\Omega$.

In particular, in the context of Example 2.8, if $\Omega$ is a closed and bounded subset of $\mathbb{R}^{N}$, we obtain the metric random walk space $[\Omega,d,m^{J,\Omega},\mathcal{L}^{N}\!\!\mathop{\hbox{ \vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega]$ where $m^{J,\Omega}:=(m^{J})^{\Omega}$; that is,

[TABLE]

for every Borel set $A\subset\Omega$ and $x\in\Omega$.

2.2. The nonlocal gradient, divergence and Laplace operators

Let us introduce the nonlocal counterparts of some classical concepts.

Definition 2.11**.**

Let $[X,\mathcal{B},m,\nu]$ be a random walk space. Given a function $f:X\rightarrow\mathbb{R}$ we define its nonlocal gradient $\nabla f:X\times X\rightarrow\mathbb{R}$ as

[TABLE]

Moreover, given ${\bf z}:X\times X\rightarrow\mathbb{R}$, its $m$-divergence ${\rm div}_{m}{\bf z}:X\rightarrow\mathbb{R}$ is defined as

[TABLE]

We define the (nonlocal) Laplace operator as follows.

Definition 2.12**.**

Let $[X,\mathcal{B},m,\nu]$ be a random walk space, we define the $m$-Laplace operator (or $m$-Laplacian) from $L^{1}(X,\nu)$ into itself as $\Delta_{m}:=M_{m}-I$, i.e.,

[TABLE]

for $f\in L^{1}(X,\nu)$. **

Note that

[TABLE]

In the case of the random walk space associated with a locally finite weighted discrete graph $G=(V,E)$ (as defined in Example 2.9), the $m^{G}$-Laplace operator coincides with the graph Laplacian (also called the normalized graph Laplacian) studied by many authors (see, for example, [4], [5], [16], [18], [24]):

[TABLE]

In [31] (see also [32]) we define and proof the following facts.

[TABLE]

and for $f\in BV_{m}(X)$ we define its $m$-total variation as

[TABLE]

For a set $E\in\mathcal{B}$ such that \raisebox{2.0pt}{\rm{\chi}}_{E}\in BV_{m}(X), we define its $m$-perimeter as

[TABLE]

If $\nu(E)<+\infty$ then

[TABLE]

The following coarea formula holds:

[TABLE]

Furthermore we give the following nonlocal concept of mean curvature. Let $E\in\mathcal{B}$ with $\nu(E)>0$. For a point $x\in X$ we define the $m$-mean curvature of $\partial E$ at $x$ as

[TABLE]

Observe that

[TABLE]

Having in mind (2.4), we have that, if $\nu(E)<+\infty$,

[TABLE]

[TABLE]

Consequently,

[TABLE]

and

[TABLE]

2.3. Schwarz’s symmetrization

Let $E\subset\mathbb{R}^{N}$ be a measurable set of finite measure, and let \raisebox{2.0pt}{\rm{\chi}}_{E} its characteristic function. The symmetric rearrangement of $E$ is the ball $E^{*}$ centered at zero with $|E^{*}|=|E|$, i.e., with radius $\left(\frac{|E|}{\omega_{N}}\right)^{\frac{1}{N}}$, where $\omega_{N}$ denotes the volume of the $N$-dimensional unit ball. For a non-negative measurable function $f:\mathbb{R}^{N}\rightarrow\mathbb{R}$ vanishing at infinity, the Schwarz’s symmetrization of $f$ is

[TABLE]

where by definition, (\raisebox{2.0pt}{\rm{\chi}}_{E})^{*}=\raisebox{2.0pt}{\rm{\chi}}_{E^{*}}. Thus, the level sets of $f^{*}$ are the rearrangements of the level sets $f$, implying the equimeasurability property

[TABLE]

The Schwarz’s symmetrization $f^{*}$ of a function $f$ inherits many measure geometric properties from its source function $f$ (see [3]). It also fulfils some optimization properties with respect to integration. We will make use of the following inequalities (see [27]), the Hardy-Littlewood’s inequality:

[TABLE]

and the Riesz’s inequality:

[TABLE]

We also need the general rearrangement inequality proved in [9]:

Theorem 2.13** (see Theorem 3.8 in [27]).**

Let $m,k\in\mathbb{N}$, $m\geq k$, and $f_{i}$, $i=1,2,...,m$, nonnegative functions in $\mathbb{R}^{N}$, vanishing at infinity. Let $B$ a $k\times m$ matrix with coefficient $b_{ij}$ in the raw $i$ and column $j$. Then, if

[TABLE]

we have that

[TABLE]

where each $f_{j}^{*}$ is the symmetric-nonincreasing rearrangement of $f_{j}$.

3. Torsional rigidity in random walk spaces

Let $[X,\mathcal{B},m,\nu]$ be a reversible random walk space. Given $\Omega\in\mathcal{B}$, we define the $m$-boundary of $\Omega$ by

[TABLE]

and its $m$-closure as

[TABLE]

From now on we will assume that $\Omega$ is $m$-connected (which imply that also $\Omega_{m}$ is $m$-connected),

[TABLE]

Remark 3.1**.**

A first consequence of the above assumptions is that

[TABLE]

Indeed, if $P_{m}(\Omega)=0$ then , by (2.4), $\displaystyle\int_{\Omega}m_{x}(\Omega)d\nu(x)=1,$ and consequently $m_{x}(\Omega)=1$ $\nu$-a.e. $x\in\Omega$. Therefore

[TABLE]

which contradicts tha $\Omega_{m}$ is $m$-connected (we are assuming $0<\nu(\Omega)<\nu(\Omega_{m})$).

On the other hand, if $P_{m}(\Omega)=\nu(\Omega)$ then, by (2.4), $m_{x}(\Omega)=0$ $\nu$-a.e. $x\in\Omega$. Therefore

[TABLE]

which contradicts that $\Omega$ is $m$-connected. $\blacksquare$

Given $p\geq 1$, we define

[TABLE]

We say that $\Omega$ satisfies a $p$-Poincaré inequality if there exists $\lambda>0$ such that

[TABLE]

for all $f\in L^{p}_{0}(\Omega_{m},\nu)$

Let us point out that the random walk spaces given in Example 2.8, for $J$ with compact support, and in Example 2.9 satisfy a $2$-Poincaré’s type inequality, see [1, 32].

In this section we will assume that $\Omega$ satisfies a $2$-Poincaré inequality.

As a consequence of the results in [45] (see also [32]), there is a unique solution of the following homogenous Dirichlet problem for the $m$-Laplacian

[TABLE]

that is,

[TABLE]

We denote by $f_{\Omega}$ this unique solution and name it as the $m$-stress function of $\Omega$. By the comparison principle given in [45], we have that $f_{\Omega}\geq 0.$

Definition 3.2**.**

The $m$-torsional rigidity of $\Omega$, $T_{m}(\Omega)$, is defined as the $L^{1}(\nu)$-norm of the torsion function:

[TABLE]

In the local case, it is well known (see, for exmaple, [7]) that

[TABLE]

Then,

[TABLE]

Contrary to the local setting, the $m$-torsional rigidity of $\Omega$ always satisfies

[TABLE]

Indeed, by the first equation in (3.3), for $x\in\Omega$, since $m_{x}(\Omega_{m})=1$, we have

[TABLE]

Hence

[TABLE]

We will give in Proposition 3.6 a detailed description of $T_{m}(\Omega)$ by using a kind of geometrical terms relative to $\Omega$ via the random walk.

The next result is the nonlocal version of equation (1.2). It is a particular case of Theorem 7.1.

Theorem 3.3**.**

We have

[TABLE]

and the maximum is attained at $f_{\Omega}$.

In [32] (see also [33]) we introduce the spectral $m$-heat content of $\Omega$ as

[TABLE]

where $v(t,x)$ is the solution of the homogeneous Dirichlet problem for the $m$-heat equation:

[TABLE]

Moreover, we have (see [32] and [33]):

[TABLE]

where, for $k\in\mathbb{N}\cup\{0\}$, $g_{m,\Omega}(k)$ is the measure of the amount of individuals that, starting in $\Omega$, end up in $\Omega$ after $k$ jumps without ever leaving $\Omega$, that is:

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

[TABLE]

i.e., $\mathbb{Q}_{\Omega}^{m}(t)$ is the expected value of the amount of individuals that start in $\Omega$ and end in $\Omega$ at time $t$ without ever leaving $\Omega$, when these individuals move by successively jumping according to $m$ and the number of jumps made up to time $t$ follows a Poisson distribution with rate $t$.

Lemma 3.4**.**

We have that

[TABLE]

Proof.

For $n\geq 1$,

[TABLE]

Then (3.11) holds. $\Box$

Remark 3.5**.**

Observe that, by (3.1), we have $g_{m,\Omega}(1)<g_{m,\Omega}(0)$. We also have

[TABLE]

Indeed, using reversibility,

[TABLE]

Then, if $g_{m,\Omega}(2)=g_{m,\Omega}(1)$, we have

[TABLE]

Hence $\Omega=A\cup B$, where $A:=\{x\in\Omega:m_{x}(\Omega)=0\}$ and up to a $\nu$-null set, $B=\{x\in\Omega:m_{x}(\Omega)=1\}.$ Now, we have

[TABLE]

and consequenlty, since $\Omega$ is $m$-connected, $\nu(A)=0$ or $\nu(B)=0$, which yields a contradiction (remember Remark 3.1). $\blacksquare$

Let us now see the nonlocal version of equation (1.10). Observe that the second statement in the next result gives a complete description of $T_{m}(\Omega)$ in term of the sequence of probabilistic terms $\{g_{m,\Omega}(n)\,:\,n\in\mathbb{N}\}$.

Theorem 3.6**.**

We have

[TABLE]

and

[TABLE]

Proof.

It is easy to see that if $v$ is the solution of the Dirichlet problem (3.8), then

[TABLE]

is the unique solution $f_{\Omega}$ of problem (3.3). Hence, by Fubini’s Theorem,

[TABLE]

By (3.12) and (3.9), since the convergence in (3.9) is uniform, we have

[TABLE]

$\Box$

As consequence of (3.13) we have the following result.

Corollary 3.7**.**

If $\Omega_{1}\subset\Omega_{2}$, then $T_{m}(\Omega_{1})\leq T_{m}(\Omega_{2})$.

Having in mind (1.9), we give the following definition.

Definition 3.8**.**

We define the sequence of exit-$m$-moments of $\Omega$ as

[TABLE]

Note that, as in (1.7),

[TABLE]

In the next result we also describe explicitly the sequence of exit-$m$-moments in terms of the sequence $\{g_{m,\Omega}(k)\,:\,k\in\mathbb{N}\}$. In the context of Riemannian manifolds, see [15] for other type of expansions.

Proposition 3.9**.**

We have

[TABLE]

Proof.

Let $j\geq 1$, then

[TABLE]

Now we can interchange the integral with the sum to get

[TABLE]

$\Box$

Let us now define

[TABLE]

Since we are assuming $\Omega$ satisfies a $2$-Poincaré type inequality, we have

[TABLE]

And, since \big{|}|a|-|b|\big{|}\leq|a-b| for all $a,b\in\mathbb{R}$,

[TABLE]

Similarly to the local case we have the following nonlocal version of (1.4) (see Corollary 6.5 later on):

[TABLE]

We also have that, see (6.25),

[TABLE]

Observe that, by (3.1) we have that $\frac{P_{m}(\Omega)}{\nu(\Omega)}<1$. Therefore, from (3.18) and  (3.17),

[TABLE]

The following assumption will be used in the next result: There exists a non-null function $f\in L^{2}(\Omega_{m},\nu)$ such that

[TABLE]

Observe that then the infimum defining $\lambda_{m,2}(\Omega)$ in (3.15) is attained at $f$. We say that $\lambda_{m,2}(\Omega)$ is the first eigenvalue of the $m$-Laplacian with homogeneous Dirichlet boundary conditions with associated eigenfunction $f$. Note that, in fact, there is a non-negative eigenfunction associated to $\lambda_{m,2}(\Omega)$.

In the next result we see that it is possible to obtain $\lambda_{m,2}(\Omega)$ via the sequence $\{g_{m,\Omega}(k)\,:\,k\in\mathbb{N}\}$ that characterize the torsional rigidity $T_{m}(\Omega)$ (Theorem 3.6) and the exit-$m$-moments (Proposition 3.9).

Theorem 3.10**.**

Assume $\lambda_{m,2}(\Omega)$ is an eigenvalue of the $m$-Laplacian with homogeneous Dirichlet boundary conditions. Then:

1. $g_{m,\Omega}(n)>0\quad\hbox{for all }n\in\mathbb{N}.$

2. Assume moreover that there exists an eigenfunction $f$ associated to $\lambda_{m,2}(\Omega)$ such that

[TABLE]

Then,

[TABLE]

Proof.

We have, for a non-negative (non-null) eigenfunction $f$ associated to $\lambda_{m,2}(\Omega)$:

[TABLE]

Now, since $0<\lambda_{m,2}(\Omega)<1$, we can write (3.23) as

[TABLE]

Then, by induction, for $n\in\mathbb{N}$,

[TABLE]

and, then, integrating over $\Omega$ with respect to $d\nu$, we have

[TABLE]

Let us see that

[TABLE]

In fact, by the reversibility of $\nu$ with respect to the random walk, for $n=1$, we have

[TABLE]

For $n=2$, using moreover Fubini’s theorem,

[TABLE]

and now we can use the case $n=1$. The general case follows by induction.

Then, by (3.24), we have

[TABLE]

therefore, $g_{m,\Omega}(n)>0.$

Proof of 2. Dividing the expression (3.24) in $n$ between the one in $n+1$, we get

[TABLE]

Therefore,

[TABLE]

or equivalently,

[TABLE]

where

[TABLE]

with $g_{m,\Omega}(n)$ given in (3.10), that is,

[TABLE]

Observe that $\tau_{n}$ is the average of $g(x_{1},x_{2},...,x_{n},x_{n+1}):=f(x_{n+1})$ in $\Omega\times...\times\Omega\times\Omega$ with respect to the measure $dm_{x_{n}}(x_{n+1})\dots dm_{x_{1}}(x_{2})d\nu(x_{1})$. Since $0<\alpha\leq f\leq\widetilde{\alpha}$, we have

[TABLE]

Now, from (3.26), we have that

[TABLE]

Hence,

[TABLE]

Since by (3.28), $\displaystyle\lim_{n}\frac{1}{n}\log\left(\frac{\tau_{2n}}{\tau_{n}}\right)=0$, taking limits in (3.30) we get (3.22). $\Box$

Remark 3.11**.**

1. Let $[\mathbb{R}^{N},d,m^{J},\mathcal{L}^{N}]$ be the metric random walk space given in Example 2.8 with $J$ continuous and compactly supported. For $\Omega$ a bounded domain, the assumption (3.21) is true, see [1, Section 2.1.1].

For weighted discrete graphs, $\lambda_{m^{G},2}(\Omega)$ is an eigenvalue with $0<\lambda_{m^{G},2}(\Omega)\leq 1$ (see [23]). Now, since we are assuming that $\Omega$ is $m^{G}$-connected, $0<\lambda_{m^{G},2}(\Omega)<1.$ And, by connectedness, using (3.23), we have that (3.21) is also true.

3. Let us see what can happen if $\Omega$ is not $m^{G}$-connected. Consider, for example, the weighted graph $G$ with five different vertices $V:=V(G)=\{x_{1},x_{2},x_{3},x_{4},x_{5}\}$ and $w_{x_{i},x_{i+1}}=1$, for $i=1,2,3,4$, and $w_{x_{i},x_{j}}=0$ otherwise. We have,

[TABLE]

[TABLE]

3.1 Take $\Omega=\{x_{2},x_{4}\}$, which is not $m^{G}$-connected. It is easy to see that $g_{m,\Omega}(n)=0$ for all $n\geq 1.$ And we have that

[TABLE]

3.2 Take now $\Omega:=\{x_{1},x_{2},x_{4},x_{5}\}$, which is also not $m^{G}$-connected. In this case $g_{m^{G},\Omega}(n)\neq 0$ for all $n\geq 1$, and

[TABLE]

Observe that $\{x_{1},x_{2}\}$ is $m^{G}$-connected, and $\{x_{4},x_{5}\}$ is also $m^{G}$-connected.

$\blacksquare$

4. The particular case of a nonlocal operator with non singular kernel

In this section we study the particular case of the random walk space given in Example 2.8, that is, we consider the metric measure space $(\mathbb{R}^{N},d,\mathcal{L}^{N})$, where $d$ is the Euclidean distance and $\mathcal{L}^{N}$ the Lebesgue measure on $\mathbb{R}^{N}$. Let $J:\mathbb{R}^{N}\to[0,+\infty[$ be a measurable, nonnegative and radially symmetric function verifying $\int_{\mathbb{R}^{N}}J(x)dx=1$. Let $m^{J}$ the random walk

[TABLE]

for which the Lebesgue measure is reversible.

We are going to prove a nonlocal version of the Saint-Venant inequality. For this we need the following result.

Lemma 4.1**.**

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$. If $J$ is radial and non-increasing, then

[TABLE]

Proof.

It is obvious that

[TABLE]

and, by Riesz inequality and having in mind that $J^{*}=J$ and (\raisebox{2.0pt}{\rm{\chi}}_{\Omega})^{*}=\raisebox{2.0pt}{\rm{\chi}}_{\Omega^{*}}, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Let us now see that

[TABLE]

Indeed, for $k=2$,

[TABLE]

Now, since

[TABLE]

choosing the $3\times 5$ matrix

[TABLE]

we have

[TABLE]

Then, by Theorem 2.13, we have

[TABLE]

The inequalities for rest of $g_{m,\Omega}$(k) are obtained similarly.

$\Box$

Theorem 4.2**.**

Let $\Omega$ be a bounded measurable subset of $\mathbb{R}^{N}$ and assume that $J$ is radial and non-increasing. Then, we have the following inequalities:

$\mathbb{Q}_{\Omega}^{m^{J}}(t)\leq\mathbb{Q}_{\Omega^{*}}^{m^{J}}(t)\quad\forall t\geq 0.$**

$T_{m^{J}}(\Omega)\leq T_{m^{J}}(\Omega^{*})$*   (Saint-Venant inequality).*

$EM^{m^{J}}_{j}(\Omega)\leq EM^{m^{J}}_{j}(\Omega^{*})\quad\forall j\geq 1.$**

Proof.

$1.$ It is consequence of (3.9) and Lemma 4.1.

$2$. It is consequence (3.13) and Lemma 4.1.

$3$. It is consequence of Proposition 3.9 and Lemma 4.1. $\Box$

Remark 4.3**.**

A Faber-Krahn inequality

[TABLE]

can be obtained as a consequence of [21, Lemma A.2]. Moreover, assuming that $J$ is decreasing, and assuming also $\lambda_{m^{J},2}(\Omega)$ is an eigenvalue, or equivalently the infimum in the Rayleigh quotient

[TABLE]

is a minimum (we know this is true for $J$ with compact support which, obviously, are not decreasing), by [21, Lemma A.2], one can also prove

[TABLE]

$\blacksquare$ **

4.1. Rescaling results

In this subsection we see that we can recover the local concepts and some of their properties from the nonlocal ones. In particular we give a different proof of the classical Saint-Venant inequality.

Set

[TABLE]

And define

[TABLE]

Observe that $C_{J_{\epsilon},2}=\frac{1}{\epsilon^{2}}C_{J,2}$.

Theorem 4.4**.**

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$. Assume $\displaystyle\int_{\mathbb{R}^{N}}J(x)|x|dx<+\infty$. We have:

[TABLE]

where $\mathbb{Q}_{\Omega}(t)$ is the (local) spectral heat content of $\Omega$; and

[TABLE]

Proof.

The first part is consequence of the rescaling results proved in [1] (see also [33]) that also work if $\int_{\mathbb{R}^{N}}J(x)|x|dx<+\infty$ thanks to the general results given by A. Ponce in [40]. The second part is a consequence of the fact that we can interchange the limit with the integral. $\Box$

By Theorems 4.4 and 4.2, we can recover the classical Saint-Venant inequality:

Theorem 4.5** (Saint-Venant inequality).**

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$. Then,

[TABLE]

And, more generally, for any $j\geq 1$,

[TABLE]

5. The particular case of a weighted graph

In this section we describe an iterative numerical method to get the torsional rigidity of a non-trivial subset of a weighted discrete graph. It is not our intention to give numerical results. We only want to show that (3.10) and (3.13) allow to use such iterative method.

Consider a weighted discrete graph $[V(G),\mathcal{B},m^{G},\nu_{G}]$ as in Example 2.9 and a $\Omega$ a finite connected subset of $V(G)$. Let us write $\Omega=\{x_{1},x_{2},...,x_{N}\}$ and $\partial_{m^{G}}\Omega=\{x_{N+1},...,x_{M}\}$, $x_{i}\neq x_{j}$ for $i\neq j$. Set $w_{ij}$ the weights between $x_{i}$ and $x_{j}$ (remember that $w_{ij}=0$ if $x_{i}\not\sim x_{j}$).

Set the weight of each $x_{i}\in\Omega$:

[TABLE]

Then, from (3.10) and (3.13), the following iterative scheme gives an approximation $T(n)$ of the torsion:

[TABLE]

From (3.26) we have that

[TABLE]

6. The $m$-$p$-torsional rigidiy

Brasco in [10], for $p>1$, defines the $p$-torsional rigidity of the set $D$ as

[TABLE]

In [10, Proposition 2.2], it is proved that

[TABLE]

where $v_{D}$ is the unique weak solution of the problem

[TABLE]

Now we are going to get the nonlocal version of equation (6.1).

In this section we will we assume that $1<p<\infty$, $\Omega\in\mathcal{B}$, $0<\nu(\Omega)<\nu(X)$ and $\Omega$ satisfies a $p$-Poincaré inequality (see (3.2)).

From the reversibility of $\nu$ respect to $m$, we have the following integration by parts formula

[TABLE]

if $f,g\in L^{p}(\Omega_{m},\nu)$.

We give the following definition of the homogeneous Dirichlet problem for the $m$-$p$-Laplacian.

Definition 6.1**.**

Given $g\in L^{1}(\Omega,\nu)$, we say that $f\in L^{p}_{0}(\Omega_{m},\nu)$ is a solution of problem

[TABLE]

if it verifies

[TABLE]

that is,

[TABLE]

Existence and uniqueness are given in [45] (see also [32]). Nevertheless, and for the sake of completeness, we give the next result with a different proof.

Theorem 6.2**.**

There is a unique solution $f_{\Omega,p}\geq 0$ of the homogenous Dirichlet problem for the $m$-$p$-Laplacian,

[TABLE]

Moreover, $f_{\Omega,p}$ is the only minimizer of the variational problem

[TABLE]

where

[TABLE]

And,

[TABLE]

Proof.

First note that $\mathcal{F}_{m,p}$ is convex and lower semicontinuous in $L^{p}(\Omega,\nu)$, thus weakly lower semicontinuous (see [11, Corollary 3.9]). Set

[TABLE]

and let $\{f_{n}\}$ be a minimizing sequence. Then,

[TABLE]

Since $\Omega_{m}$ satisfies a the Poincaré inequality (3.2), by Young’s inequality, we have

[TABLE]

[TABLE]

[TABLE]

Therefore, we obtain that

[TABLE]

Hence, up to a subsequence, we have

[TABLE]

Furthermore, using the weak lower semicontinuity of the functional $\mathcal{F}_{m,p}$, we get

[TABLE]

Since the functional $\mathcal{F}_{m,p}$ is strictly convex, we have that $f_{\Omega,p}$ is the unique minimizer, and since $\mathcal{F}_{m,p}(|f|)\leq\mathcal{F}_{m,p}(f)$, we have that $f_{\Omega,p}\geq 0$.

Thus, given $\lambda>0$ and $w\in L_{0}^{p}(\Omega,\nu)$ , we have

[TABLE]

or, equivalently,

[TABLE]

[TABLE]

Now, since $p>1$, we pass to the limit as $\lambda\downarrow 0$ to obtain

[TABLE]

Taking $\lambda<0$ and proceeding as above we obtain the opposite inequality. Consequently, we conclude that

[TABLE]

[TABLE]

which shows that $f_{\Omega,p}$ is solution of (6.5).

Finally, taking $w=f_{\Omega,p}$ in the above first equation we get (6.7). $\Box$

Definition 6.3**.**

We call to $f_{\Omega,p}$ as the $p$-torsional function of $\Omega$, and we define the $m$-$p$-torsional rigidity of $\Omega$ as

[TABLE]

Note that $T_{m}(\Omega)=T_{m,2}(\Omega)$. **

Theorem 6.4**.**

We have

[TABLE]

and the maximum is attained at $f_{\Omega,p}$.

Proof.

By (6.7),

[TABLE]

Therefore,

[TABLE]

Let $g\in L_{0}^{p}(\Omega_{m},\nu)$, $g\not=0$. Since $f_{\Omega,p}$ is a solution of Problem (6.5),

[TABLE]

Then, by Hölder’s inequality,

[TABLE]

Then, from (6.7),

[TABLE]

[TABLE]

Thus,

[TABLE]

and consequently (6.8) holds. $\Box$

We now define, for $p\geq 1$,

[TABLE]

As a consequence of the above result we have:

Corollary 6.5**.**

For $p>1$ we have

[TABLE]

Proof.

By Theorem 7.1, we have

[TABLE]

Now

[TABLE]

Hence

[TABLE]

$\Box$

Fusco, Maggi and Pratelli in [22] (see also [2], [19] and [41]) generalized the classical concept of Cheeger constant, introducing, for $p\geq\frac{N-1}{N}$, the $p$-Cheeger constant of and open set $\Omega\subset\mathbb{R}^{N}$ of finite measure as

[TABLE]

Note that for $h_{1}(\Omega)$ is the classical Cheeger constant.

In [31] (see also [32]), for a set $\Omega\in\mathcal{B}$ such that $0<\nu(\Omega)<\nu(X)$, we define its $m$-Cheeger constant as

[TABLE]

and we prove (see [32, Theorem 3.37]) that

[TABLE]

Remark 6.6**.**

For any $p\geq 1$,

[TABLE]

Indeed, for any $E\subset\Omega$, $\nu(E)>0$,

[TABLE]

[TABLE]

Then taking infimum, and on account of (6.12), we get (6.13). $\blacksquare$

Now, we introduce the following nonlocal version of the $p$-Cheeger constant.

Definition 6.7**.**

Let $p>1$, we define its $m$-$p$-Cheeger constant of $\Omega$ as

[TABLE]

Similarly to the local case (see, for example, [13, Proposition 5.2]), we have the following relation between the Chegeer constants and the $m$-$p$-torsional rigidity.

Theorem 6.8**.**

For $p>1$ we have

[TABLE]

and

[TABLE]

Proof.

By the coarea formula (2.5) and Cavalieri’s principle, we have

[TABLE]

[TABLE]

[TABLE]

Hence, by Hölder’s inequality and (6.7), we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and, from here

[TABLE]

On the other hand, by (6.8), for any $E\in\mathcal{B},\ E\subset\Omega,\ \nu(E)>0$, we have

[TABLE]

from where,

[TABLE]

And (6.14) is proved.

Taking limits in (6.14), we have

[TABLE]

and

[TABLE]

Let us now see that

[TABLE]

Indeed, for any $E\in\mathcal{B},\ E\subset\Omega,\ \nu(E)>0$, we have

[TABLE]

and, from here

[TABLE]

which allows to prove (6.19). Finally, (6.18) and (6.19) gives (6.15). $\Box$

Pólya [39] proves that, among all bounded open and convex planar sets, the following inequality holds

[TABLE]

being the constant $\frac{1}{3}$ optimal. This was generalized in [12] to dimension $N\geq 3$. On the other hand, Makai [28] proves that, among all bounded open and convex planar sets, the following upper bound holds

[TABLE]

being the constant $\frac{2}{3}$ optimal. See  [12] for a conjecture in dimension $N\geq 3$. Estimates (6.20) and (6.21) are generalized for the $p$-Laplacian by Fragala, Gazzola and Lamboley in [20].

Recall that $\Omega$ is $m$-calibrable if $h_{1}^{m}(\Omega)=\frac{P_{m}(\Omega)}{\nu(\Omega)}.$

Corollary 6.9**.**

We have

[TABLE]

Moreover, if $\Omega$ is $m$-calibrable, then

[TABLE]

Proof.

Taking $p=2$ in (6.14), since $T_{m,2}(\Omega)=T_{m}(\Omega)$, we have

[TABLE]

Then, since $h_{2}^{m}(\Omega)\leq\frac{P_{m}(\Omega)}{\nu(\Omega)^{2}}$, from the second inequality in (6.24) we get

[TABLE]

and (6.22) holds. On the other hand, assuming that $\Omega$ is $m$-calibrable, we have $h_{1}^{m}(\Omega)=\frac{P_{m}(\Omega)}{\nu(\Omega)}$, and, substituting this value in the first inequality of (6.24), we have

[TABLE]

from where (6.23) holds. $\Box$

Observe that, from (3.1), (3.17) and (6.22), we have

[TABLE]

In the next example we will see that the second and third inequalities in (6.25) are sharp. We see that they are equalities for the most simple connected set for weighted discrete graphs, which is trivially $m^{G}$-calibrable.

Example 6.10**.**

1. Consider the weighted discrete lasso graph $V(G)=\{x,y\}$ with weights $w_{xx}=a>0$, $w_{xy}=b>0$ and $w_{yy}=0$ (we are in a situation of Example 2.9). And take $\Omega=\{x\}$, which is $m^{G}$-connected (because of the loop). It is easy to see that

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Hence,

[TABLE]

2. For the weighted discrete graph $V(G)=\{x,y_{1},y_{2},\dots,y_{k}\}$, $k\geq 2$ with weights $w_{xx}=a>0$, $w_{xy_{i}}=b_{i}>0$ and $w_{y_{i}y_{j}}=0$ for any $i,j$, if we set $b=\sum_{i=1}^{k}b_{j}$, and take $\Omega=\{x\}$, we have the same results than for the lasso graph. $\blacksquare$

In the next result we will see the influence of the $m$-mean curvature of $\Omega$. Observe first that, by (2.9),

[TABLE]

Then, (6.22) is equivalent to

[TABLE]

Remember also that

[TABLE]

Then, as an inmediate consequence of (6.27) we have:

Corollary 6.11**.**

Assume that b there exists $\beta\in\mathbb{R}$ such thatb

[TABLE]

Then

[TABLE]

By (2.9), we have

[TABLE]

Now, since (6.22) can be written as

[TABLE]

we obtain the following result.

Corollary 6.12**.**

Assume that there exists $\alpha\in\mathbb{R}$ such that

[TABLE]

Then

[TABLE]

Remark 6.13**.**

1. Let us remark that, assuming (6.30), by the above Corollary and by (3.17), we have

[TABLE]

2. Observe that (6.29) is a Pólya-type inequality for subsets satisfying (6.28); and that (6.23) is a Makai-type inequality for calibrable subsets.

3. As a consequence of (6.27) and (6.23), if $\Omega$ is calibrable then

[TABLE]

or equivalently, using (2.9),

[TABLE]

$\blacksquare$

We have the following result (see [26] in the local case).

Theorem 6.14**.**

We have,

[TABLE]

And consequently,

[TABLE]

Proof.

The second inequality of (6.31) is given in (6.13). On the other hand, for $p>1$, we have, for any $a,b\in\mathbb{R}$,

[TABLE]

Hence

[TABLE]

and consequently, for $u\in L_{0}^{p}(\Omega_{m},\nu)\setminus\{0\}$, we have

[TABLE]

We claim now that

[TABLE]

Indeed, by the reversibility of $\nu$ respect to $m$, and having in mind that $\nabla u(x,y)=0$ if $u(x)=u(y)$ and $|\nabla u(x,y)|=|\nabla u(y,x)|$, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Now, applying Hölder’s inequality, we get

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where reversibility is used, as in the proof of (6.34), to get the last equality. Then

[TABLE]

Hence, using the above inequality, from (6.33) we get

[TABLE]

[TABLE]

Thus

[TABLE]

The, taking infimum in $u\in L_{0}^{p}(\Omega_{m},\nu)\setminus\{0\}$, we obtain that

[TABLE]

and (6.31) is proved. Finally, (6.32) is a direct consequence of (6.31) and (6.12). $\Box$

6.1. A rescaling result

Set $J_{\epsilon}$ as in (4.3). Define

[TABLE]

Observe that $C_{J_{\epsilon}}=\frac{1}{\epsilon}C_{J}$.

If $\displaystyle\int_{\mathbb{R}^{N}}J(x)|x|dx<+\infty$, we have (see [33]):

[TABLE]

Remember that by (4.4),

[TABLE]

Now, from (6.24),

[TABLE]

Then

[TABLE]

and, taking limits as $\epsilon\to 0$,

[TABLE]

Observe that

[TABLE]

But, $\frac{C_{J}{}^{2}}{2C_{J,2}}$ is as close to 1 as we want by choosing adequately $J$. So we can get

[TABLE]

and in particular, for $\Omega$ calibrable we get the Makai-type inequality

[TABLE]

7. Torsional rigidity on Quantum Graphs as a $m$-torsional rigidity on graphs

Torsional rigidity on quantum graphs was introduce by Colladay, Kaganovskiy and McDonald in [14]. To the best of our knowledge, after this paper, the only existing literature on this topic is the paper by Mugnolo and Plumer [35], where the torsional rigidity of a quantum graph is related to the rigidity of an associated weighted combinatorial graph. We will interpret here that result with the (nonlocal) rigidity of a weighted graph.

Let $\mathcal{G}$ be a compact, finite, connected quantum graph. Let $V$ be the set of vertices of $\mathcal{G}$ and $E$ be the set of edges. Fora vertex $x\in V$, let ${\rm deg}_{\mathcal{G}}(x)$ denote is degree, i.e. the number of edges incident in $x$. We suppose that $\mathcal{G}$ has at least one vertex of degree $1$. Set

[TABLE]

and set $V_{N}:=V\setminus V_{D}$. We assume that the graph does not contain multiple edges between the same vertices but it can contain at most one loop at each vertex (we comment on this later on). Let us call $\ell_{e}$ or $\ell_{x,y}$ the length of the edge $e$ that join the vertices $x$ and $y$.

For each $e\in E$ there exists an increasing an bijective function

[TABLE]

$x_{e}$ is called the coordinate of the point $x\in e$.

A function $u$ on a metric graph $\mathcal{G}$ is a collection of functions $[u]_{e}$ defined on $[0,\ell_{e}]$ for all $e\in E.$ Throughout this work, $\int_{\mathcal{G}}u(x)dx$ denotes $\sum_{e\in E}\int_{0}^{\ell_{e}}[u]_{e}(x_{e})\,dx_{e}$.

For $A\subset\mathcal{G}$, the length of $A$ is defined as

[TABLE]

Let $\Delta_{\mathcal{G}}$ the Laplacian on $\mathcal{G}$ with homogeneous Dirichlet boundary condition at vertices in $V_{D}$ and with the Kirchhoff type condition on the vertices in $V_{N}$, that is, its associated quadratic form $a_{\mathcal{G}}$ is given by

[TABLE]

on the domain

[TABLE]

Let $v$ be the solution of

[TABLE]

The function $v$ is called the torsion function of $\mathcal{G}$, and the (quantum) torsional rigidity of $V_{D}$ is given by the $L^{1}$-norm of $v$:

[TABLE]

In [35] Mugnolo and Plumer show that, if $v$ is the torsion function of $\mathcal{G}$, then $f=2v_{|_{V}}:V\rightarrow\mathbb{R}$ is the unique solution of the following problem:

[TABLE]

And they prove that

[TABLE]

Observe that in the above expression, $\displaystyle\sum_{y\sim x}\ell_{yx}+\ell_{xx}=\sum_{y\sim x,\,y\neq x}\ell_{yx}+2\ell_{xx}$. If we had $k(\geq 2)$ loops at the vertex $x$ with lengths $\ell_{xx}(i)$, $i=1,2,...k$, then we should change $\displaystyle 2\ell_{xx}$ by $2(\ell_{xx}(1)+...+\ell_{xx}(k)).$

Take $c>0$ large enough such that (we do not mark the dependence on $c$)

[TABLE]

Observe that, since $\mathcal{G}$ is finite, such a $c$ exists.

Let us consider the weighted graph $G_{c}$ having the same vertices and edges than $\mathcal{G}$ with weights (we do not mark the dependence on $c$ in $\widetilde{w}_{xx}$):

[TABLE]

On account of (7.4), we have that

[TABLE]

And, then, from (7.2), we have that $f_{c}:=2c^{2}v_{|_{V}}$ satisfies

[TABLE]

Observe that, since $V_{D}=\partial_{m^{G_{c}}}V_{N}$, $f_{c}$ is solution of the problem

[TABLE]

Then we have that formula (7.3) given in [35] can be written using weighted discrete graphs, seen as random walk spaces, as follows.

Theorem 7.1**.**

We have

[TABLE]

whatever $c$ is chosen in (7.4).

Proof.

Indeed, from (7.5),

[TABLE]

And hence the statement (7.7) follows from (7.3). $\Box$

As a consequence of the above theorem and (6.22) we recover the equivalent to Proposition 4.8 of [35].

Corollary 7.2**.**

We have, for any $c>0$ satisfying (7.4),

[TABLE]

Remark 7.3**.**

1. Observe that if we assume that $\ell_{e}=1$ for all edge $e$ in $\mathcal{G}$, and we have not loops,

[TABLE]

Indeed, $\nu_{G_{c}}(V_{N})=c\sum_{x\in V_{N}}{\rm deg}_{\mathcal{G}}(x)$ and $P_{m^{G_{c}}}(V_{N})=\frac{1}{c}\sum_{x\in V_{N}}\sharp(\{y\in V_{D}:y\sim x\}).$ Then, the first inequality in (7.9) follows from (7.8), and the second inequality follows since, for each $x\in V_{N}$, ${\rm deg}_{\mathcal{G}}(x)\geq\sharp(\{y\in V_{D}:y\sim x\})$.

2. Consider a star metric graph $\mathcal{G}$, with Dirichlet conditions imposed on all vertices except the central one, and with a possible loop in the central vertex. Suppose that there are $k$ Dirichlet vertices with their edges joining the central vertex having length $\ell_{i}$, $i=1,2,...,k$, and the possible loop at the central vertex with length $\ell_{0}\geq 0$ (if $\ell_{0}=0$ we do not have a loop and we have only a star). Then, on account of Theorem 7.1 and Example 6.10, for $c$ satisfying (7.4), we have that

[TABLE]

The above equality recover, as could not be otherwise, the result of Example 3.10 of [35]. We see that in this case that we have equality in (7.8) (this is also remarked in [35, Proposition 4.8]). In the particular case that $\ell_{i}=1$ for $i=1,2,...,k$ and $\ell_{0}=0$, then $T_{q}(\mathcal{G})=\frac{1}{3}k$, and all the inequalities in (7.9) are equalities. $\blacksquare$

Acknowledgments. The authors have been partially supported by Conselleria d’Innovació, Universitats, Ciència y Societat Digital, project AICO/2021/223.