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.
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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
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
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
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
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
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]
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
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
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:
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.
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
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
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
and
![](http://media.springernature.com/lw262/springer-static/image/art%3A10.1007%2Fs00208-024-02847-3/MediaObjects/208_2024_2847_Equ92_HTML.png)
for some \(f_i \in L^1(\Omega )\). So, defining
we can write
![](http://media.springernature.com/lw181/springer-static/image/art%3A10.1007%2Fs00208-024-02847-3/MediaObjects/208_2024_2847_Equ93_HTML.png)
Clearly it holds
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
Moreover if \(u \in L^1(\Omega )\) and
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
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
and the increasing rearrangement of u as
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
where \(\omega _n\) is the Lebesgue measure of the unit n-dimensional ball.
Moreover the spherically symmetric increasing rearrangement of u is defined as
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
Let \(v \in W_0^{1,p}(\Omega )\) be a function that satisfies
denoting by \(z \in W_0^{1,p}(\Omega ^{\sharp })\) the unique spherically decreasing symmetric solution to
then
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
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
and u a non-negative function belonging to \(\text {BV}_0(\Omega )\). Let us define
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
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. \(s \in [0,+\infty )\):
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
Moreover, we have
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, \(\forall s\ge 0\)
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
3 Proof of Theorem 1.2
Let us define the following function
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
Proof
The isoperimetric inequality implies
Hence, for all \(0 \le s_1< s_2 < + \infty \) we have
where
Since this holds for every open interval \((s_1,s_2)\), we have
Observing that H is a Lipschitz function, \(D\big [H(u^*)\big ]\) is given by (see [5])
since \(\mu (u^*(s))= s\) a.e. with respect \(Du^*\) (by the properties of the rearrangements) and since for \(s \in J_{u^*}\)
Then we can write
Therefore, by means of (3.3), (3.4), we have
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
Now, let us integrate (3.2) between 0 and \(+\infty \) and let us use Fubini’s Theorem to obtain
By (2.9) applied to \(F_1\) and the Hardy-Littlewood inequality (2.2), we have
then
since \(F_2(A)=0\) for all \(A \subset ({\left|\Omega \right|},+\infty )\).
Using again Fubini’s Theorem, we can compute
and
Hence, (3.5) can be written as
Remark 3.1
We stress the following facts:
and then
4 Two versions of the torsional rigidity
For a given \(\Lambda >0\) we consider
and the associated minimum problem:
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
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)$$\begin{aligned} -T_\mathcal {F}(B_R, \Lambda )=\liminf _k \mathcal {F}_{\Lambda }(\psi _k) \ge \mathcal {F}_{\Lambda }(v) \ge -T_\mathcal {F}(B_R,\Lambda ) \end{aligned}$$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
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
and then
Now we deal with the functional
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
It is easy to check that the Euler-Lagrange equation of this functional is
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
Proof
For every function \(\psi \in H^{1}(\Omega )\), by 1.2, there exists \({\overline{\psi }} \in H^{1}(\Omega ^{\sharp })\) that satisfies
and then
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We would like to thank the anonymous Referees for their valuable comments and suggestions.
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Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement. This work has been partially supported by the PRIN project (Italy) Grant: “Direct and inverse problems for partial differential equations: theoretical aspects and applications” and by GNAMPA of INdAM.
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