On the gradient rearrangement of functions

Abstract

In this paper, we introduce a symmetrization technique for the gradient of a $\text {BV}$ function, which separates its absolutely continuous part from its singular part (sum of jump and Cantorian part). In particular, we prove a $\text {L }^{\text {1}}$ comparison between the function and the symmetrization just mentioned. Furthermore, we apply this result to obtain Saint-Venant type inequalities for some geometric functionals.

1 Introduction

Let $\Omega$ be a bounded open set of $\mathbb {R}^n$ with finite perimeter (see Sect. 2 for its definition) and let us denote, as in [7], by

$$\begin{aligned} \text {BV}_0(\Omega ):= \left\{ u \in \text {BV}(\mathbb {R}^n) :\, u\equiv 0 \text { in }\mathbb {R}^n \setminus \Omega \right\} . \end{aligned}$$

The aim of the present paper is to define a symmetrization of the distributional gradient of a BV function.

The interest in this topic essentially derives from the work [17] where the authors deal with the following problems involving Hamilton-Jacobi equation

$$\begin{aligned} {\left\{ \begin{array}{ll} H(\nabla u) = f &{} \text {a.e. in } \Omega \\ u = 0 &{} \text {on } \partial \Omega \end{array}\right. } \end{aligned}$$

(1.1a)

$$\begin{aligned} {\left\{ \begin{array}{ll} K({\left|\nabla v\right|}) = f_{\sharp } &{} \text {a.e. in } \Omega ^{\sharp } \\ v = 0 &{} \text {on } \partial \Omega ^{\sharp } \end{array}\right. } \end{aligned}$$

(1.1b)

where $\Omega ^{\sharp }$ is the ball centered at the origin with the same measure as $\Omega$ (in the sequel just centered ball), $H: \mathbb {R}^n \rightarrow \mathbb {R}$ and $K:\mathbb {R}\rightarrow \mathbb {R}$ are measurable functions, $u,v \in W^{1,p}_0$ and $f_{\sharp }$ is the increasing rearrangement of f (see Sect. 2 for its definition).

In particular, under suitable assumptions on H and K, it is proven ([17, Theorem 2.2]) that whenever u, v are solutions to (1.1a) and (1.1b) respectively, then

$$\begin{aligned} {\left\Vert u\right\Vert }_{L^1(\Omega )} \le {\left\Vert v\right\Vert }_{L^1(\Omega ^{\sharp })}. \end{aligned}$$

In [2] the authors study the problem of maximization of the $L^q$ norm among functions with prescribed gradient rearrangement. Precisely, the following cases are considered

• $1 \le q \le \frac{np}{n-p}$ if $p<n$,

• $1 \le q <+\infty$ if $p = n$,

• $1 \le q \le + \infty$ if $p>n$,

and for a fixed $\varphi = \varphi ^{*} \in L^p(0,{\left|\Omega \right|})$, they define

$$\begin{aligned} I(\Omega ) : = \sup \left\{ {\left\Vert v\right\Vert }_{L^q} \, : \; \; \begin{aligned}&{\left|\nabla v\right|} \le f \; \text {a.e. in }\Omega , \\&v \in W_0^{1,p}(\Omega ) \\&f \ge 0, \, f^* = \varphi ^* \end{aligned} \right\} , \end{aligned}$$

and they proved the following

Theorem 1.1

[2, Theorem 3.1] Let $\Omega$ be a bounded open set in $\mathbb {R}^n$, let $\Omega ^{\sharp }$ be the centered ball, let R be its radius and let $p,q,\varphi$ be as defined above.

Then, there exist v, g spherically symmetric on $\Omega ^{\sharp }$ such that $g^* = \varphi$, $I(\Omega ^{\sharp }) = {\left\Vert v\right\Vert }_{L^q}$, and thus

$$\begin{aligned} v \in W_0^{1,p}(\Omega ), \, v \ge 0, \, {\left|\nabla v\right|} = g \qquad \text {a.e. in }\Omega ^{\sharp }. \end{aligned}$$

Furthermore $I(\Omega ^{\sharp }) \ge I(\Omega )$ for all open sets $\Omega$ in $\mathbb {R}^n$ with $|\Omega ^{\sharp } |= {\left|\Omega \right|}$.

In [9] the author proved a representation formula for the function g, the existence of which was proved in Theorem 1.1.

Let us also mention that in [14, 15] the authors studied the optimization of the norm of a Sobolev function in the class of functions with prescribed rearrangement of the gradient.

The case of a Sobolev non-zero trace function for $q=1$ is instead studied in [4].

The literature concerning rearrangements in the spaces $W^{1,p}$ is exhaustive, whereas, to our knowledge, results on $\text {BV}$ functions are rarer. One of the most relevant papers in this framework is [10] where the authors extend the validity of Polya-Szegö inequality to $\text {BV}$ functions. More specifically, they proved that if $u \in \text {BV}(\mathbb {R}^n)$, then its Schwarz rearrangement $u^{\sharp }$ (see Sect. 2 for its definition) belongs to $\text {BV}(\mathbb {R}^n)$ and it holds [10, Theorem 1.3]

$$\begin{aligned} \begin{aligned} |D u^{\sharp } |(\mathbb {R}^n)&\le |D u |(\mathbb {R}^n) \\ |D^{\textrm{s}} u^{\sharp } |(\mathbb {R}^n)&\le |D^{\textrm{s}} u |(\mathbb {R}^n) \\ |D^{\textrm{j}} u^{\sharp } |(\mathbb {R}^n)&\le |D^{\textrm{j}} u |(\mathbb {R}^n) \end{aligned} \end{aligned}$$

(1.2)

where $D^s$ and $D^j$ denote respectively the singular and the jump part of the gradient (see [10] for their definitions). There is no analogue of (1.2) for the absolutely continuous and the Cantorian part of the gradient, i.e. in the symmetrization procedure the total variation of $D^a$ and $D^c$ can be increased, as shown in the example given in [10].

In this paper, we want to introduce a symmetrization that keeps the absolutely continuous part separate from the singular part (sum of jump and Cantorian part) of the gradient. To be more precise, we define the radial function $u^{\star } \in W^{1,1}(\Omega ^{\sharp })\cap \text {BV}_0(\Omega ^{\sharp }) \cap L^{\infty }(\Omega ^\sharp )$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} |\nabla u^{\star } |(x) = {\left|\nabla ^a u\right|}_{\sharp }(x) &{} \text {a.e. in }\Omega ^{\sharp } \\[1ex] u^\star (x)= \frac{1}{\displaystyle { \text {Per}(\Omega ^\sharp )}} |D^s u |(\mathbb {R}^n) &{}\text { on } \partial \Omega ^{\sharp } \end{array}\right. }, \end{aligned}$$

(1.3)

where $\nabla ^a u$ and $D^s u$ will be defined in Sect. 2.

The main theorem can be stated as follows.

Theorem 1.2

Let $\Omega \subset \mathbb {R}^n$ be a bounded open set with finite perimeter and let $\Omega ^\sharp$ be the centered ball. Assume that u is a non-negative function belonging to $\text {BV}_0(\Omega )$ and assume that $u^\star$ is defined as in (1.3), then

$$\begin{aligned} {\left\Vert u\right\Vert }_{L^1(\Omega )} \le {\left\Vert u^{\star }\right\Vert }_{L^1(\Omega ^{\sharp })}. \end{aligned}$$

We will also deal with some applications, in particular we will consider

• a penalized torsional rigidity problem

  $$\begin{aligned} T_\mathcal {F}(\Omega ,\Lambda ) := - \inf _{\psi \in H_0^1(\Omega )} \biggl ( \frac{1}{2} \int _{\Omega } {\left|\nabla \psi \right|}^2 \, dx - \int _{\Omega } {\left|\psi \right|} \, dx + \Lambda {\left| \left\{ {\left|\nabla \psi \right|} \ne 0 \right\} \right|} \biggr ); \end{aligned}$$

• a modified torsional rigidity

  $$\begin{aligned} \frac{1}{T_\mathcal {G}(\Omega ,m)} := \inf _{\psi \in H^1(\Omega )} \frac{\displaystyle {\int _{\Omega } {\left|\nabla \psi \right|}^2 \, dx + \frac{1}{m}\left( \int _{\partial \Omega } {\left|\psi \right|} \, d\mathcal {H}^{n-1}\right) ^2}}{\displaystyle {\left( \int _{\Omega } {\left|\psi \right|} \, dx\right) ^2}}. \end{aligned}$$

In both cases, we will prove a Saint-Venant type inequality:

$$\begin{aligned} T_{\mathcal {F}}(\Omega ,\Lambda ) \le T_{\mathcal {F}}(\Omega ^{\sharp }, \Lambda ), \qquad T_{\mathcal {G}}(\Omega ,m) \le T_{\mathcal {G}}(\Omega ^{\sharp },m). \end{aligned}$$

The paper is organized as follows: in Sect. 2 we recall some preliminary results and useful tools for our aim, in Sect. 3 we prove our main result on the symmetrization of the gradient for a BV function, while in Sect. 4 we present some applications of this kind of symmetrization.

2 Notations and preliminaries

2.1 Functions of bounded variation

Let us summarize some basic notions concerning $\text {BV}$ functions, for all the details we refer for instance to [6, 10, 13].

In the following, $\Omega$ will be an open set of $\mathbb {R}^n$.

Definition 2.1

A function $u \in L^1(\Omega )$ is said to be a function of bounded variation in $\Omega$ if its distributional derivative is a Radon measure, i.e.

$$\begin{aligned} \int _{\Omega } u {\frac{\partial \varphi }{\partial x_i}} \, dx = \int _{\Omega } \varphi \, dD^i u \qquad \forall \varphi \in C_C^{\infty }(\Omega ), \end{aligned}$$

with Du a $\mathbb {R}^n$-valued measure in $\Omega$. The total variation of Du will be denoted with ${\left|Du\right|}$.

The set of functions of bounded variation in $\Omega$ is denoted by $\text {BV}(\Omega )$ and it is a Banach space with respect to the norm ${\left\Vert u\right\Vert }_{\text {BV}(\Omega )}: = {\left\Vert u\right\Vert }_{L^1(\Omega )} + {\left|Du\right|}(\Omega )$.

Definition 2.2

Let E be a $\mathcal {L}^n$-measurable set. The perimeter of E inside $\Omega$ is defined as

$$\begin{aligned} \text {Per}(E,\Omega ) : = {\left|D \chi _E\right|}(\Omega ), \end{aligned}$$

and we say that E is a set of finite perimeter in $\Omega$ if $\chi _E \in \text {BV}(\Omega )$. If $\Omega = \mathbb {R}^n$, we denote $\text {Per}(E):=\text {Per}(E,\mathbb {R}^n)$.

It is also worth mentioning the isoperimetric inequality for sets of finite perimeter.

Theorem 2.1

(Isoperimetric inequality) Let $E\subset \mathbb {R}^n$ be a bounded set of finite measure. Then it holds

$$\begin{aligned} {\left|E\right|} \le n^{-\frac{n}{n-1}}\omega _n^{-\frac{1}{n-1}}\left[ \text {Per}(E)\right] ^{\frac{n}{n-1}}, \end{aligned}$$

where $\omega _n$ is the measure of n-dimensional ball of radius 1.

By the Lebesgue decomposition Theorem, each component of Du can be decomposed with respect to the Lebesgue measure, namely

$$\begin{aligned} D_i u = D_i^{\textrm{a}}u + D_i^{\textrm{s}}u \qquad \text { with } D_i^{\textrm{a}}u \ll \mathcal {L}^n, \quad D_i^{\textrm{s}}u \perp \mathcal {L}^n. \end{aligned}$$

and

for some $f_i \in L^1(\Omega )$. So, defining

$$\begin{aligned} \frac{\partial u}{\partial x_i} : = f_i, \qquad \nabla ^{\textrm{a}} u = \biggl ( \frac{\partial u}{\partial x_1},\cdots , \frac{\partial u}{\partial x_n} \biggr ) \qquad \text { and } D^{\textrm{s}} u = \bigl ( D_1^{\textrm{s}}u, \ldots , D_n^{\textrm{s}}u \bigr ), \end{aligned}$$

we can write

Clearly it holds

$$\begin{aligned} {\left|Du\right|}(A) = |D^{\textrm{a}} u |(A) + |D^{\textrm{s}} u |(A) = \int _{A} {\left|\nabla ^{\textrm{a}} u\right|} \, dx + |D^{\textrm{s}} u |(A), \end{aligned}$$

for every Borel set $A \subseteq \Omega$.

Let us recall the following Fleming-Rishel formula (see [16] or [13]):

Theorem 2.2

(Fleming-Rishel formula) Let $\Omega \subset \mathbb {R}^n$ be an open set and let $u \in \text {BV}(\Omega )$, then for almost every $t \in (-\infty ,+\infty )$ the set $\{ u>t\}$ has finite perimeter in $\Omega$ and it holds

$$\begin{aligned} {\left|Du\right|}(\Omega ) = \int _{-\infty }^{+\infty } \text {Per}(\{ u>t\}, \Omega ) \, dt. \end{aligned}$$

(2.1)

Moreover if $u \in L^1(\Omega )$ and

$$\begin{aligned} \int _{-\infty }^{+\infty } \text {Per} (\{ u > t \}, \Omega ) \, dt < +\infty , \end{aligned}$$

then $u \in \text {BV}(\Omega )$ and consequently (2.1) holds.

2.2 Rearrangements of functions

We now briefly recall some notions about rearrangements. We refer for instance to [18, 19, 23] for all the details.

Definition 2.3

Let $\Omega$ be a measurable set and let $u :\Omega \rightarrow \mathbb {R}$ be a measurable function, the distribution function of u is defined as

$$\begin{aligned} \mu :[0,+\infty ) \rightarrow [0,+\infty ) \qquad \mu (t) = \left|\bigl ( \left\{ x \in \Omega \, : {\left|u(x)\right|}>t\right\} \bigr ) \right|\end{aligned}$$

where, here and throughout the paper, ${\left|E\right|}$ denotes the n-dimensional Lebesgue measure of a measurable set E.

It can be proved that

• $\mu$ is a decreasing function in $[0,+\infty )$;

• $\mu$ is right-continuous;

• $\mu (0) = {\left| \text {supp}u \right|}$ and $\mu (+\infty ) = 0$;

• $\mu (t^-) = \bigl |\left\{ x \in \Omega : \, {\left|u(x)\right|} \ge t \right\} \bigr |$.

Definition 2.4

Let $u :\Omega \rightarrow \mathbb {R}$ be a measurable function, the decreasing rearrangement of u is defined as

$$\begin{aligned} u^* :\mathbb {R}^+ \rightarrow \mathbb {R}^+ \qquad u^*(s) = \inf \left\{ t>0 \, : \mu (t) \le s\right\} \end{aligned}$$

and the increasing rearrangement of u as

$$\begin{aligned} u_* :[0,{\left|\Omega \right|}] \rightarrow \mathbb {R}^+ \qquad u_*(s) = u^*({\left|\Omega \right|}-s) \end{aligned}$$

It can be proved that

• $u^*$ ($u_*$) is a decreasing (increasing) function in $[0,+\infty )$;

• $u^*$ and $u_*$ are lower semi-continuous;

• whenever $u \in L^{\infty }(\Omega )$ $u^*(0) = {\left\Vert u\right\Vert }_{L^{\infty }(\Omega )}$ and $u^*(t) = 0$ $\forall t \ge {\left| \text {supp}u \right|}$;

• $u_*({\left|\Omega \right|}) = {\left\Vert u\right\Vert }_{L^{\infty }(\Omega )}$ and $u_*(t) = 0$ $\forall 0 \le t \le {\left|\Omega \right|} -{\left| \text {supp}u \right|}$;

• $u^*$ and $u_*$ have the same distribution function as u, so by Cavalieri’s principle the $L^p$ norms are equal for every p;

• $u^*(\mu (t)) \le t$ for every non-negative t, $\mu (u^*(s)) \le s$ for every non-negative s;

• $u^*(\mu (t)^-) \ge t$ for every non-negative t, $\mu (u^*(s)^-) \ge s$ for every non-negative s;

• the Hardy-Littlewood inequality: for any $u,v :\Omega \subseteq \mathbb {R}^n \rightarrow \mathbb {R}$

  $$\begin{aligned} \int _{\Omega } {\left|u(x)v(x)\right|} \, dx \le \int _{\Omega ^{\sharp }} u^*(x) v^*(x) \, dx = \int _{\Omega } u_*(x) v_*(x) \, dx \end{aligned}$$

  (2.2)

Definition 2.5

Let $u :\Omega \rightarrow \mathbb {R}$ be a measurable function. The Schwarz rearrangement or the spherically symmetric decreasing rearrangement of u is defined as

$$\begin{aligned} u^{\sharp } :\mathbb {R}^n \rightarrow \mathbb {R}^+ \qquad u^{\sharp }(x) = u^*(\omega _n {\left|x\right|}^n) \end{aligned}$$

where $\omega _n$ is the Lebesgue measure of the unit n-dimensional ball.

Moreover the spherically symmetric increasing rearrangement of u is defined as

$$\begin{aligned} u_{\sharp } :\mathbb {R}^n \rightarrow \mathbb {R}^+ \qquad u_{\sharp } (x) = u_*(\omega _n {\left|x\right|}^n) \end{aligned}$$

It can be proved that

• $u^{\sharp }$ ($u_{\sharp }$) is non-negative, radial and radially decreasing (increasing);

• $u^{\sharp }, u_{\sharp }$ and u are equally distributed which means they have the same distribution function;

• the Polya-Szegö inequality holds true [21]: if $u \in W_0^{1,p}(\Omega )$, then $u^{\sharp } \in W_0^{1,p}(\Omega ^{\sharp })$ and

  $$\begin{aligned} \Vert \nabla u^{\sharp } \Vert _{L^p(\Omega ^{\sharp })} \le {\left\Vert \nabla u\right\Vert }_{L^p(\Omega )}. \end{aligned}$$

We recall the Theorem of Giarrusso and Nunziante ([17, Theorem 2.2]).

Theorem 2.3

Let $\Omega \subset \mathbb {R}^n$ be a bounded open set, let $\Omega ^{\sharp }$ be the centered ball, let $p \ge 1$, let $f :\Omega \rightarrow \mathbb {R}$ be a measurable function, let $H :\mathbb {R}^n \rightarrow \mathbb {R}$ be measurable non-negative functions and let $K :[0,+\infty ) \rightarrow [0,+\infty )$ be a strictly increasing real-valued function such that

$$\begin{aligned} 0 \le K({\left|y\right|}) \le H(y) \qquad \forall y \in \mathbb {R}^n \qquad \text { and } K^{-1}(f) \in L^p(\Omega ). \end{aligned}$$

Let $v \in W_0^{1,p}(\Omega )$ be a function that satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} H(\nabla v) = f(x) &{}\text {a.e. in }\Omega \\ v = 0 &{}\text {on } \partial \Omega \end{array}\right. }, \end{aligned}$$

denoting by $z \in W_0^{1,p}(\Omega ^{\sharp })$ the unique spherically decreasing symmetric solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} K({\left|\nabla z\right|}) = f_{\sharp }(x) &{} \text {a.e. in } \Omega ^{\sharp } \\ z = 0 &{} \text {on } \partial \Omega ^{\sharp } \end{array}\right. }, \end{aligned}$$

then

$$\begin{aligned} {\left\Vert v\right\Vert }_{L^1(\Omega )} \le {\left\Vert z\right\Vert }_{L^1(\Omega ^{\sharp })}. \end{aligned}$$

Moreover, in [20] the following uniqueness result is proved:

Theorem 2.4

Let $\Omega \subset \mathbb {R}^n$ be a bounded open set, let $v \in W_0^{1,1}(\Omega )$ be a non-negative function. Denote by $f(x) = {\left|\nabla v\right|}(x)$ and by $w \in W_0^{1,1}(\Omega ^{\sharp })$ the decreasing spherically symmetric solution to

$$\begin{aligned} {\left|\nabla w\right|} = f_{\sharp }. \end{aligned}$$

If ${\left\Vert v\right\Vert }_{L^1} = {\left\Vert w\right\Vert }_{L^1}$ then there exists $x_0 \in \mathbb {R}^n$ such that $\Omega = x_0 + \Omega ^{\sharp }$, $f=f_{\sharp }(\cdot \, + x_0)$ and $v = w (\cdot \, + x_0)$.

From now on $\Omega \subset \mathbb {R}^n$ is a bounded open set with finite perimeter. Let us consider

$$\begin{aligned} \text {BV}_0(\Omega ):= \left\{ u \in \text {BV}(\mathbb {R}^n) :\, u\equiv 0 \text { in }\mathbb {R}^n \setminus \Omega \right\} , \end{aligned}$$

and u a non-negative function belonging to $\text {BV}_0(\Omega )$. Let us define

$$\begin{aligned} f(x,s) = \bigl ( u - u^*(s)\bigr )_+(x) \qquad x \in \mathbb {R}^n, \, s \in [0,+\infty ). \end{aligned}$$

(2.3)

The function $f(\cdot , s)$ belongs to $\text {BV}_0(\Omega )$ for every $s \in [0, +\infty )$ since it is a truncation of u (See [6, Theorem 3.96]). Moreover, for every $s \in [0,+\infty )$ we denote by

$$\begin{aligned} G(s) = |D f(\cdot ,s) |(\mathbb {R}^n) = |D^a f(\cdot ,s) |(\mathbb {R}^n) + |D^s f(\cdot ,s) |(\mathbb {R}^n) = G_1(s) + G_2(s), \end{aligned}$$

(2.4)

where $D^a f$ and $D^s f$ are, respectively, the absolutely continuous part and singular part of the measure Df.

The following corollary holds.

Corollary 2.5

Let u be a non-negative function belonging to $\text {BV}_0(\Omega )$ and let G(s) be the function defined as in (2.4). Then for a.e. $s \in [0,+\infty )$:

$$\begin{aligned} G(s)= \int _{u^*(s)}^{+\infty } \text {Per}(\{ u>\xi \} ) \, d\xi . \end{aligned}$$

(2.5)

Proof

For a.e. $s \in [0,+\infty )$, applying 2.2 with $E=\mathbb {R}^n$ to the function $f(\cdot ,s)$ defined in (2.3), we have

$$\begin{aligned} G(s)={\left|D \bigl ( (u-u^*(s) \bigr )_+)\right|}(\mathbb {R}^n) = \int _{-\infty }^{+\infty } \text {Per}\left( \left\{ \bigl ( u - u^*(s)\bigr )_+> \xi \right\} \right) \, d\xi . \end{aligned}$$

(2.6)

Moreover, we have

$$\begin{aligned} \int _{-\infty }^{+\infty } \text {Per}\left( \left\{ \bigl ( u - u^*(s)\bigr )_+> \xi \right\} \right) \, d\xi = \int _{0}^{+\infty } \text {Per}\left( \left\{ u - u^*(s)> \xi \right\} \right) \, d\xi , \end{aligned}$$

and a change of variables gives (2.5).

The following properties hold:

• 1. G is an increasing function on $(0,+\infty )$ by (2.5), constant in $({\left|\Omega \right|}, +\infty )$, it belongs to $\text {BV}_{\text {loc}}([0,+\infty ))$. Then, there exists a positive measure $\sigma$ such that

  $$\begin{aligned} G(s)= \int _{(0,s]} \, d\sigma (\tau ) \qquad \forall s \in [0,+\infty ); \end{aligned}$$

  (2.7)

• 2. $\displaystyle {G_1(s) = \int _{\{u > u^*(s)\}} |\nabla ^{\textrm{a}} u |\, dx}$ is increasing and AC on $[0,+\infty )$, then there exists a function $F_1$ belonging to $L^1([0,+\infty ))$:

  $$\begin{aligned} G_1(s)= \int _0^s F_1(\tau )\, d\tau \qquad \forall s \in [0,+\infty ); \end{aligned}$$

• 3. $G_2$ is an increasing function belonging to $\text {BV}_{\text {loc}}([0,+\infty ))$, so there exists a positive measure $\sigma _2$ such that

  $$\begin{aligned} G_2(s)= \int _{(0,s]}\, d\sigma _2(\tau ) \qquad \forall s \in [0,+\infty ). \end{aligned}$$

Then, $\forall s\ge 0$

$$\begin{aligned} G(s) =\sigma ((0,s])= \int _{(0,s]} \, d\sigma (\tau ) = \int _{0}^s F_1(\tau ) \,d\tau + \int _{(0,s]} d\sigma _2(\tau ) \end{aligned}$$

(2.8)

We will need the following technical lemma which can be proved by arguing as [3, Lemma 2.1].

Lemma 2.6

Let $\Omega$ be a bounded open set in $\mathbb {R}^n$. If $g \in L^1([0,{\left|\Omega \right|}))$, then there exists a sequence of functions $\{g_k\}$ such that $g_k^* = g^*$ and

$$\begin{aligned} \lim _k \int _0^{{\left|\Omega \right|}} g_k (s) \varphi (s) \, dx = \int _0^{{\left|\Omega \right|}} g(s) \varphi (s) \, ds, \qquad \forall \varphi \in \text {BV}\bigl ( [0,{\left|\Omega \right|}) \bigr ). \end{aligned}$$

(2.9)

3 Proof of Theorem 1.2

Let us define the following function

$$\begin{aligned} v(s) := \int _s^{+\infty }\frac{1}{ n \omega _n^{\frac{1}{n}} \tau ^{1-\frac{1}{n}}} \, d\sigma (\tau )\qquad \forall s \in [0,+\infty ), \end{aligned}$$

(3.1)

where $\sigma$ is defined in (2.7). We observe that, since $\text {supp}( \sigma ) \subseteq [0,{\left|\Omega \right|}]$, v is identically 0 on $({\left|\Omega \right|}, +\infty )$, hence $v \in \text {BV}_0([0,{\left|\Omega \right|}])$.

As intermediate step towards Theorem 1.2, we prove the following proposition.

Proposition 3.1

Let $\Omega \subset \mathbb {R}^n$ be a bounded open set with finite perimeter and assume that u is a non-negative function belonging to $\text {BV}_0(\Omega )$. If v(s) is the function defined as in (3.1), then

$$\begin{aligned} u^*(s) \le v(s) \qquad \text { for a.e. } s \in [0,+\infty ). \end{aligned}$$

(3.2)

Proof

The isoperimetric inequality implies

$$\begin{aligned} n \omega _n^{\frac{1}{n}} \mu (t)^{1-\frac{1}{n}} \le \text {Per}(\{u>t\}) \qquad \forall t \in [0.+\infty ), \end{aligned}$$

by (2.5) and (2.8) we have

$$\begin{aligned} G(s) = \int _{u^*(s)}^{+\infty } \text {Per}(\left\{ u> \xi \right\} ) \, d\xi = \int _{(0,s]}\, d\sigma (\tau ) \qquad \text { for a.e. }s \in [0,+\infty ). \end{aligned}$$

Hence, for all $0 \le s_1< s_2 < + \infty$ we have

$$\begin{aligned} \sigma \big ((s_1,s_2)\big )=\int _{s_1}^{s_2} \, d\sigma (\tau )&= \lim _{s \rightarrow s_2^-}G(s) - G(s_1) \\ {}&=\lim _{s \rightarrow s_2^-} \int _{u^\star (s)}^{u^\star (s_1)}\text {Per}(\left\{ u> \xi \right\} ) \, d\xi \\ {}&\ge \lim _{s \rightarrow s_2^-}\int _{u^\star (s)}^{u^\star (s_1)} n \omega _n^{\frac{1}{n}} \mu (\xi )^{1-\frac{1}{n}} \, d\xi = D\big [H(u^*)\big ]\big ((s_1,s_2)\big ), \end{aligned}$$

where

$$\begin{aligned} H(\tau ) = \int _{\tau }^{+\infty } n \omega _n^{\frac{1}{n}} \mu (\xi )^{1-\frac{1}{n}} \, d\xi . \end{aligned}$$

Since this holds for every open interval $(s_1,s_2)$, we have

$$\begin{aligned} \sigma (A) \ge D\big [H(u^*)\big ](A) \qquad \forall A \subseteq [0, +\infty )\text { Borel set}. \end{aligned}$$

(3.3)

Observing that H is a Lipschitz function, $D\big [H(u^*)\big ]$ is given by (see [5])

$$\begin{aligned} D\big [H(u^*)\big ] = {\left\{ \begin{array}{ll} -n \omega _n^{\frac{1}{n}} s^{1-\frac{1}{n}} Du^*&{} \text { on } [0, +\infty ) \setminus J_{u^*} \\ - n \omega _n^{\frac{1}{n}} s^{1-\frac{1}{n}}\left( (u^*)^+- (u^*)^-\right) ,&{} \text { on } J_{u^*} \end{array}\right. } \end{aligned}$$

since $\mu (u^*(s))= s$ a.e. with respect $Du^*$ (by the properties of the rearrangements) and since for $s \in J_{u^*}$

$$\begin{aligned} H\bigl ( ( (u^*)^+(s) \bigr )- H \bigl ( ( (u^*)^-(s)\bigr )&= \int _{ u^*(s)}^{u^*(s^-)} n \omega _n^{\frac{1}{n}} \mu (\xi )^{1-\frac{1}{n}} \, d\xi \\&= - n \omega _n^{\frac{1}{n}} s^{1-\frac{1}{n}}\left( (u^*)^+(s)- (u^*)^-(s)\right) . \end{aligned}$$

Then we can write

$$\begin{aligned} \frac{dD\big [H(u^*)\big ] }{dDu^*}= - n \omega _n^{\frac{1}{n}} s^{1-\frac{1}{n}}. \end{aligned}$$

(3.4)

Therefore, by means of (3.3), (3.4), we have

$$\begin{aligned} u^*(s) = -\int _s^{+\infty } \, d(D u^*)(\tau ) = \int _s^{+\infty } \frac{dD\big [H(u^*)\big ](\tau )}{n \omega _n^{ \frac{1}{n} }\tau ^{1 - \frac{1}{n}} }\le \int _s^{+\infty } \frac{d\sigma (\tau )}{n \omega _n^{ \frac{1}{n} } \tau ^{1 - \frac{1}{n}} }= v(s). \end{aligned}$$

Now we are in position to prove the main theorem.

Proof of Theorem 1.2

First of all, let us emphasize that the decreasing rearrangement of $u^\star$, defined in (1.3), is

$$\begin{aligned} (u^\star )^*(s)= \int _s^{+\infty } \frac{ |\nabla ^{\textrm{a}} u |_* (t) }{n\omega _n^{\frac{1}{n}}t^{1-\frac{1}{n}}} \, dt + \frac{1}{\text {Per}(\Omega ^\sharp )} {\left|D^s u\right|}(\mathbb {R}^n) \, \chi _{[0,{\left|\Omega \right|}]}(s) \qquad \forall s \in [0,+\infty ). \end{aligned}$$

Now, let us integrate (3.2) between 0 and $+\infty$ and let us use Fubini’s Theorem to obtain

$$\begin{aligned} \int _0^{+\infty } u^*(s) \, ds&\le \int _0^{+\infty } v(s) \, ds \\&= \frac{1}{n \omega _n^{\frac{1}{n}}} \int _0^{+\infty } \Biggl ( \int _s^{+\infty } \, \frac{d\sigma (t)}{t^{1-\frac{1}{n}}} \Biggr ) \, ds \\&= \frac{1}{n\omega _n^{\frac{1}{n}}} \int _0^{+\infty } \Biggl ( \int _0^t \, \frac{ds}{t^{1-\frac{1}{n}}} \Biggr ) \, d\sigma (t) \\&= \frac{1}{n\omega _n^{\frac{1}{n}}} \int _0^{+\infty } t^{\frac{1}{n}} \, d \sigma (t)\\&=\frac{1}{n\omega _n^{\frac{1}{n}}} \left[ \int _0^{+\infty }t^{\frac{1}{n}} F_1(t) \, dt + \int _0^{+\infty } t^{\frac{1}{n}} \, d\sigma _2(t)\right] . \end{aligned}$$

By (2.9) applied to $F_1$ and the Hardy-Littlewood inequality (2.2), we have

$$\begin{aligned} \int _0^{+\infty } t^{\frac{1}{n}} F_1(t) \, dt&=\int _0^{{\left|\Omega \right|}} t^{\frac{1}{n}} F_1(t) \, dt = \lim _k \int _0^{{\left|\Omega \right|}} t^{\frac{1}{n}} (F_1)_k(t) \, dt \\&\le \int _0^{{\left|\Omega \right|}} t^{\frac{1}{n}} |\nabla ^{\textrm{a}} u |_* (t) \, dt = \int _0^{+\infty } t^{\frac{1}{n}} |\nabla ^{\textrm{a}} u |_* (t) \, dt, \end{aligned}$$

then

$$\begin{aligned} \begin{aligned} \int _0^{+\infty } u^*(s) \, ds&\le \frac{1}{n\omega _n^{\frac{1}{n}}} \Biggl [ \int _0^{+\infty } t^{\frac{1}{n}} |\nabla ^{\textrm{a}} u |_* (t) \, dt + \int _0^{+\infty } t^{\frac{1}{n}} \, d\sigma _2(t) \Biggr ] \\&\le \frac{1}{n\omega _n^{\frac{1}{n}}} \Biggl [ \int _0^{+\infty } t^{\frac{1}{n}} |\nabla ^{\textrm{a}} u |_* (t) \, dt + {\left|\Omega \right|}^{\frac{1}{n}} \int _0^{+\infty } \, d\sigma _2(t) \Biggr ], \end{aligned} \end{aligned}$$

(3.5)

since $F_2(A)=0$ for all $A \subset ({\left|\Omega \right|},+\infty )$.

Using again Fubini’s Theorem, we can compute

$$\begin{aligned} \int _0^{+\infty } |\nabla ^{\textrm{a}} u |_* (t) t^{\frac{1}{n}} \, dt&= \int _0^{+\infty } \frac{ |\nabla ^{\textrm{a}} u |_* (t) }{t^{1-\frac{1}{n}}} \int _0^t \, ds = \int _0^{+\infty } \Biggl ( \int _s^{+\infty } \frac{ |\nabla ^{\textrm{a}} u |_* (t) }{t^{1-\frac{1}{n}}} \, dt \Biggr ) \, ds, \end{aligned}$$

and

$$\begin{aligned} \frac{{\left|\Omega \right|}^{\frac{1}{n}}}{n \omega _n^\frac{1}{n}} \int _0^{+\infty } dF_2(t){} & {} = {\left|\Omega \right|} \frac{1}{\text {Per}(\Omega ^\sharp )} {\left|D^s u\right|}(\mathbb {R}^n) \\{} & {} = \int _0^{+\infty }\frac{1}{\text {Per}(\Omega ^\sharp )} {\left|D^s u\right|}(\mathbb {R}^n) \chi _{[0,{\left|\Omega \right|}]}(s) \,ds. \end{aligned}$$

Hence, (3.5) can be written as

$$\begin{aligned} {\left\Vert u\right\Vert }_{L^1(\Omega )}&\le \int _0^{+\infty }\Biggl [ \int _s^{+\infty } \frac{ |\nabla ^{\textrm{a}} u |_* (t) }{n\omega _n^{\frac{1}{n}}t^{1-\frac{1}{n}}} \, dt + \frac{1}{\text {Per}(\Omega ^\sharp )} {\left|D^s u\right|}(\mathbb {R}^n) \chi _{[0,{\left|\Omega \right|}]}(s)\Biggr ]\, ds \\ {}&={\left\Vert u^\star \right\Vert }_{L^1(\Omega ^{\sharp })}. \end{aligned}$$

Remark 3.1

We stress the following facts:

$$\begin{aligned} {\left|D^{\textrm{a}} u\right|}(\mathbb {R}^n) = \int _{\mathbb {R}^n} {\left|\nabla ^\textrm{a} u\right|} \, dx = \int _{\Omega ^{\sharp }} {\left|\nabla ^\textrm{a} u^{\star }\right|} \, dx \quad \text { and } \quad {\left|D^{s} u\right|}(\mathbb {R}^n) = {\left|D^{s} u^\star \right|}(\mathbb {R}^n), \end{aligned}$$

and then

$$\begin{aligned} {\left|Du\right|}(\mathbb {R}^n)={\left|Du^\star \right|}(\mathbb {R}^n). \end{aligned}$$

4 Two versions of the torsional rigidity

For a given $\Lambda >0$ we consider

$$\begin{aligned} \mathcal {F}_{\Lambda }(\psi ) := \frac{1}{2} \int _{\Omega } {\left|\nabla \psi \right|}^2 \, dx - \int _{\Omega } \psi \, dx + \Lambda {\left| \left\{ {\left|\nabla \psi \right|} \ne 0 \right\} \right|} \qquad \psi \in H_0^1(\Omega ), \end{aligned}$$

(4.1)

and the associated minimum problem:

$$\begin{aligned} T_\mathcal {F}(\Omega ,\Lambda ) := - \inf _{\psi \in H_0^1(\Omega )} \mathcal {F}_{\Lambda }(\psi ). \end{aligned}$$

(4.2)

First of all, let us observe that the minimum can be found among non-negative functions. Indeed, passing from $\psi$ to ${\left|\psi \right|}$ it holds $\mathcal {F}(\psi ) \ge \mathcal {F}({\left|\psi \right|})$.

Assuming that problem (4.2) admits a minimum $u \in H^1_0(\Omega )$, then it is also a maximum for the torsional rigidity defined by Diaz, Polya and Weinstein in [12, 22] of a multiply-connected cross-section with fixed measure of the holes, that is

$$\begin{aligned} T(\Omega ) = \max _{\begin{array}{c} \psi \in C_0(D) \cap C^1(\Omega ) \\ \psi \text { constant} \\ \text {in every }A_i \end{array}} \frac{\displaystyle {\biggl (\int _{D} \psi \, dx \biggr )^2}}{\displaystyle {\int _{D} {\left|\nabla \psi \right|}^2 \, dx}}, \end{aligned}$$

where $A_i$ are the connected component of $\{{\left|\nabla u \right|}=0\}$ and $D = \Omega \cup \bigcup _i A_i$.

Functionals with penalizing terms are very common in the mathematical modelling of physical problems. The bibliography is very wide and some cornerstones are [1, 11].

However, in the literature, penalizing terms of the form ${\left|\left\{ {\left|\nabla \psi \right|} \ne 0\right\} \right|}$ are quite unusual. The main difficulty in the study of (4.2) is to prove the existence of a minimizer because of the lack of the lower semicontinuity of the functional.

For this reason, we prove the existence of a minimizer in the case when $\Omega$ is a ball.

Proposition 4.1

Let $\Lambda , R>0$ and let $B_R$ be the centered ball with radius R. Then the functional $\mathcal {F}_\Lambda$ defined in (4.1) admits a minimizer v belonging to $H_0^1(\Omega )$. Such a minimizer is unique up to a sign, it is radially symmetric and ${\left|\nabla v\right|}$ is radially increasing.

Proof

We divide the proof in 3 steps.

• 1. Boundness from below.

  First of all, let us prove that the functional $\mathcal {F}_{\Lambda }$ is bounded from below for every choice of $\Lambda$ and for every $R>0$. For all $\psi \in H^1_0(B_R)$, sing Young and Poincaré inequalities, we get

  $$\begin{aligned} \mathcal {F}_{\Lambda }(\psi )&= \frac{1}{2} \int _{B_R} {\left|\nabla \psi \right|}^2 \, dx - \int _{B_R} \psi \, dx + \Lambda {\left| \left\{ \nabla \psi \ne 0 \right\} \right|} \\&\ge \frac{1}{2} \int _{B_R} {\left|\nabla \psi \right|}^2 \, dx - \varepsilon \int _{B_R} \frac{\psi ^2}{2} - \frac{{\left|B_R\right|}}{2\varepsilon } \\&\ge \frac{1}{2} \int _{B_R} {\left|\nabla \psi \right|}^2 \, dx - \frac{\varepsilon C(n,B_R)}{2} \int _{B_R} {\left|\nabla \psi \right|}^2 \, dx - \frac{{\left|B_R\right|}}{2\varepsilon } \\&= \frac{(1-\varepsilon C(n,B_R))}{2} \int _{B_R} {\left|\nabla \psi \right|}^2 \, dx - \frac{{\left|B_R\right|}}{2 \varepsilon }. \end{aligned}$$

  Chosing $\varepsilon$ sufficiently small such that

  $$\begin{aligned} 0< \varepsilon \le \frac{1}{C(n,B_R)} \end{aligned}$$

  then

  $$\begin{aligned} \mathcal {F}_{\Lambda }(\psi ) \ge - \frac{{\left|B_R\right|} }{2 C(n,B_r)} \ge - C (n,B_R) > -\infty \end{aligned}$$

  so

  $$\begin{aligned} T(B_R, \Lambda ) = -\inf _{\psi \in H_0^1(B_R)} \mathcal {F}_{\Lambda }(\psi ) <\infty . \end{aligned}$$

• 2. Compactness and semicontinuity.

  Now we consider a minimizing sequence $\{\psi _k\}$ for $T_\mathcal {F}(B_R,\Lambda )$ and we prove that it is bounded in $H_0^1(B_R)$. We can assume that $\mathcal {F}_{\Lambda }(\psi _k) \le -T_\mathcal {F}(B_R,\Lambda )+1$ and by Proposition 2.3 we can assume that $\psi _k$ are radial function with ${\left|\nabla \psi _k\right|}$ radially symmetric increasing.

  Using Young and Poincaré inequalities, we obtain

  $$\begin{aligned} \mathcal {F}_{\Lambda }(\psi _k)&= \frac{1}{2} \int _{B_R} {\left|\nabla \psi _k\right|}^2 \, dx - \int _{B_R} \psi _k \, dx + \Lambda {\left| \left\{ \nabla \psi _k \ne 0 \right\} \right|} \\&\ge \frac{1}{2} \int _{B_R} {\left|\nabla \psi _k\right|}^2 \, dx - \int _{B_R} \psi _k \, dx \\&\ge \frac{1}{2} \int _{B_R} {\left|\nabla \psi _k\right|}^2 \, dx - \varepsilon \int _{B_R} \frac{\psi _k^2}{2} - \frac{{\left|B_R\right|}}{2\varepsilon } \\&\ge \frac{1}{2} \int _{B_R} {\left|\nabla \psi _k\right|}^2 \, dx - \frac{\varepsilon C(n,B_r)}{2} \int _{B_R} {\left|\nabla \psi _k\right|}^2 \, dx - \frac{{\left|B_R\right|}}{2\varepsilon } \\&= \frac{1-\varepsilon C(n,B_R)}{2} \int _{B_R} {\left|\nabla \psi _k\right|}^2 \, dx - \frac{{\left|B_R\right|}}{2\varepsilon }. \end{aligned}$$

  Choosing $\varepsilon <\displaystyle { \frac{1}{C(n,B_R) }}$ we have

  $$\begin{aligned} -T_\mathcal {F}(B_R,\Lambda )+1 \ge \mathcal {F}_{\Lambda }(\psi _k) \ge \frac{1}{4} \int _{B_R} {\left|\nabla \psi _k\right|}^2 \, dx - C (B_R) \end{aligned}$$

  then by Poincaré inequality, the sequence $\{\psi _k\}$ is bounded in $H_0^1(B_R)$.

  This implies that there exists a subsequence (still denoted by $\psi _k$) and a function $v \in H_0^1(B_R)$ such that $\psi _k \rightarrow v$ strongly in $L^2(\Omega )$, a.e. in $\Omega$ and $\nabla \psi _k \rightharpoonup \nabla v$ weakly in $L^2$. Let us show that v is a minimum for $\mathcal {F}_{\Lambda }$. The lower semicontinuity of the norms gives

  $$\begin{aligned} \liminf _k \biggl [ \frac{1}{2} \int _{B_R} {\left|\nabla \psi _k\right|}^2 \, dx - \int _{B_R} \psi _k \, dx \biggr ] \ge \frac{1}{2} \int _{B_R} {\left|\nabla v\right|}^2 \, dx - \int _{B_R} v \, dx.\nonumber \\ \end{aligned}$$

  (4.3)

  Let us deal with the last term of $\mathcal {F}_{\Lambda }$ and let us prove that

  $$\begin{aligned} \liminf _k {\left| \left\{ {\left|\nabla u_k\right|} \ne 0 \right\} \right|} \ge {\left|\left\{ {\left|\nabla u\right|} \ne 0 \right\} \right|}. \end{aligned}$$

  Denoting by $r_k$ the radius of the ball where ${\left|\nabla \psi _k\right|}=0$, we can assume that $r_k$ converges to some $r \ge 0$. Therefore

  $$\begin{aligned} \liminf _k \, {\left| \left\{ {\left|\nabla \psi _k\right|} \ne 0\right\} \right|} = \lim _k \, [\omega _n (R^n-r_k^n)] = \omega _n (R^n - r^n ). \end{aligned}$$

  So we have just to prove that ${\left|\nabla v\right|} = 0$ in $B_r$. Since $\{ \psi _k \}$ are radial functions, obviously v is radial too. If $r=0$ there is nothing to prove. If $r > 0$, assume by contradiction that there exists $A \subset B_r$ with ${\left|A\right|}>0$ and that ${\left|\nabla v\right|} \ne 0$ in A. Clearly there exists $\varepsilon >0$ such that ${\left|A \cap B_{r-\varepsilon }\right|} >0$.

  Since $r_k \rightarrow r$ if we choose a function $g \in C^{\infty }_C(B_R, \mathbb {R}^n)$ with support included in $A \cap B_{r-\varepsilon }$ we have

  $$\begin{aligned} \int _{B_R} \langle \nabla v , g \rangle \, dx = \lim _k \int _{B_R} \langle \nabla \psi _k , g \rangle \, dx = 0. \end{aligned}$$

  Since this must be true for every $g \in C^{\infty }_{C}(A \cap B_{r-\varepsilon },\mathbb {R}^n)$, we get a contradiction.

  Then in any case

  $$\begin{aligned} \liminf _k {\left|\left\{ {\left|\nabla \psi _k\right|} \ne 0\right\} \right|} \ge {\left|\left\{ {\left|\nabla v\right|} \ne 0\right\} \right|}. \end{aligned}$$

  (4.4)

  By (4.3) and(4.4), we get

  $$\begin{aligned} -T_\mathcal {F}(B_R, \Lambda )=\liminf _k \mathcal {F}_{\Lambda }(\psi _k) \ge \mathcal {F}_{\Lambda }(v) \ge -T_\mathcal {F}(B_R,\Lambda ) \end{aligned}$$

  so v is a minimum of $\mathcal {F}_{\Lambda }$ in $B_R$.

• 3. Uniqueness.

  Let us suppose that v is a minimum of $\mathcal {F}_\Lambda (\psi )$. By Theorem 2.3, it exists $\overline{v} \in H_0^1(B_R)$ such that

  $$\begin{aligned} \mathcal {F}_{\Lambda } (v) \ge \mathcal {F}_{\Lambda }(\overline{v}) \end{aligned}$$

  and since v is minimum, it holds

  $$\begin{aligned} \mathcal {F}_{\Lambda } (v) = \mathcal {F}_{\Lambda }(\overline{v}). \end{aligned}$$

  Since ${\left|\nabla v\right|}$ is equally distributed with ${\left|\nabla \overline{v}\right|}$, the previous equality implies

  $$\begin{aligned} {\left\Vert v\right\Vert }_{L^1} = {\left\Vert \overline{v}\right\Vert }_{L^1} \end{aligned}$$

  so Theorem 2.4 gives that ${\left|v\right|} = \overline{v}$.

Remark 4.1

We highlight that Theorem 2.3 ensures us that the minimum when $\Omega$ is a ball has gradient equal to zero only in a ball $B_r$ centered at the origin with $0 \le r \le R.$

Now, as already mention in the introduction, we prove a Saint-Venant type inequality for $T_\mathcal {F}(\Omega ,\Lambda )$.

Corollary 4.2

Let $\Omega \subset \mathbb {R}^n$ be a bounded open set with finite perimeter and let $\Omega ^{\sharp }$ be the centered ball. If $\Lambda > 0$, then

$$\begin{aligned} T_\mathcal {F}(\Omega ,\Lambda ) \le T_\mathcal {F}(\Omega ^{\sharp }, \Lambda ). \end{aligned}$$

Proof

For every function $\psi \in H_0^{1}(\Omega )$, by Theorem 2.3 or 1.2, there exists ${\overline{\psi }} \in H_0^{1}(\Omega ^{\sharp })$ that satisfies

$$\begin{aligned} \mathcal {F}_{\Lambda }(\psi ) \ge \mathcal {F}_{\Lambda }({\overline{\psi }}) \ge -T_\mathcal {F}(\Omega ^{\sharp },\Lambda ) \end{aligned}$$

and then

$$\begin{aligned} T_\mathcal {F}(\Omega , \Lambda ) \le T_\mathcal {F}(\Omega ^{\sharp },\Lambda ). \end{aligned}$$

Now we deal with the functional

$$\begin{aligned} \mathcal {G}(\psi ) := \frac{\displaystyle {\int _{\Omega } {\left|\nabla \psi \right|}^2 \, dx + \frac{1}{m}\left( \int _{\partial \Omega } {\left|\psi \right|} \, d\mathcal {H}^{n-1}\right) ^2}}{\displaystyle {\left( \int _{\Omega } {\left|\psi \right|} \, dx\right) ^2}} \qquad \psi \in H^1(\Omega ). \end{aligned}$$

with $m > 0$.

The interest in this type of functional is related to the problem of optimal insulation in a given domain. Indeed, the minimum of $\mathcal {G}$ gives the long-time distribution of temperature of the domain $\Omega$ and the displacement around $\Omega$ of a thin layer of insulator with total mass equal to m. We refer to [8] for more details.

If $\Omega$ is a Lipschitz domain, $\mathcal {G}(\psi )$ achieves its minimum among all $H^1(\Omega )$ functions. So we define

$$\begin{aligned} \frac{1}{T_\mathcal {G}(\Omega ,m)}:= \min _{\psi \in H^1(\Omega )} \mathcal {G}(\psi ). \end{aligned}$$

It is easy to check that the Euler-Lagrange equation of this functional is

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u= 1 &{} \text {in }\Omega \\ \displaystyle {\frac{\partial u}{\partial \nu } + \frac{1}{m} \int _{\partial \Omega } {\left|u\right|} \, d\mathcal {H}^{n-1}} = 0 &{}\text { on } \partial \Omega . \end{array}\right. } \end{aligned}$$

So Theorem 1.2 gives us the following Saint-Venant type inequality for $T_{\mathcal {G}}(\Omega )$.

Corollary 4.3

Let $\Omega \subset \mathbb {R}^n$ be a bounded open set with finite perimeter and let $\Omega ^{\sharp }$ be the centered ball. If $m > 0$, then

$$\begin{aligned} T_\mathcal {G}(\Omega ,m) \le T_\mathcal {G}(\Omega ^{\sharp },m). \end{aligned}$$

Proof

For every function $\psi \in H^{1}(\Omega )$, by 1.2, there exists ${\overline{\psi }} \in H^{1}(\Omega ^{\sharp })$ that satisfies

$$\begin{aligned} \mathcal {G}(\psi ) \ge \mathcal {G}({\overline{\psi }}) \ge \frac{1}{T_\mathcal {G}(\Omega ^{\sharp },m)} \end{aligned}$$

and then

$$\begin{aligned} T_\mathcal {G}(\Omega ,m) \le T_\mathcal {G}(\Omega ^{\sharp },m). \end{aligned}$$