On the gradient rearrangement of functions

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


Introduction
Let Ω be a bounded open set of R n with finite perimeter (see section 2 for its definition) and let us denote, as in [BDNT15], by BV 0 (Ω) := {u ∈ BV(R n ) : u ≡ 0 in R n \Ω} .
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 [GN84] where the authors deal with the following problems involving Hamilton-Jacobi equation where Ω ♯ is the ball centered at the origin with the same measure as Ω (in the sequel just centered ball), H : R n → R and K : R → R are measurable functions, u, v ∈ W 1,p 0 and f ♯ is the increasing rearrangement of f (see Section 2 for its definition).
In particular, under suitable assumptions on H and K, it is proven ([GN84, Theorem 2.2]) that whenever u, v are solutions to (1.1a) and (1.1b) respectively, then In [ALT89] the authors study the problem of maximization of the L q norm among functions with prescribed gradient rearrangement.Precisely, the following cases are considered and for a fixed ϕ = ϕ * ∈ L p (0, |Ω|), they define , and they proved the following Theorem 1.1.[ALT89,Theorem 3.1] Let Ω be a bounded open set in R n , let Ω ♯ be the centered ball, let R be its radius and let p, q, ϕ be as defined above.
In [Cia96] 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 [FP91;FPV93] 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 [AG22].
The literature concerning rearrangements in the spaces W 1,p is exhaustive, whereas, to our knowledge, results on BV functions are rarer.One of the most relevant papers in this framework is [CF02] where the authors extend the validity of Polya-Szegö inequality to BV functions.More specifically, they proved that if u ∈ BV(R n ), then its Schwarz rearrangement u ♯ (see Section 2 for its definition) belongs to BV(R n ) and it holds [CF02, Theorem 1.3] where D s and D j denote respectively the singular and the jump part of the gradient (see [CF02] 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 [CF02].
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 where ∇ a u and D s u will be defined in Section 2.
The main theorem can be stated as follows.
Theorem 1.2.Let Ω ⊂ R n be a bounded open set with finite perimeter and let Ω ♯ be the centered ball.Assume that u is a non-negative function belonging to BV 0 (Ω) and assume that u ⋆ is defined as in ) .We will also deal with some applications, in particular we will consider • a penalized torsional rigidity problem In both cases, we will prove a Saint-Venant type inequality: The paper is organized as follows: in Section 2 we recall some preliminary results and useful tools for our aim, in Section 3 we prove our main result on the symmetrization of the gradient for a BV function, while in Section 4 we present some applications of this kind of symmetrization.

Functions of bounded variation
Let us summarize some basic notions concerning BV functions, for all the details we refer for instance to [AFP00; CF02; EG15].
In the following, Ω will be an open set of R n .
Definition 2.1.A function u ∈ L 1 (Ω) is said to be a function of bounded variation in Ω if its distributional derivative is a Radon measure, i.e.

ˆΩ u ∂ϕ ∂x
with Du a R n -valued measure in Ω.The total variation of Du will be denoted with |Du|.
The set of functions of bounded variation in Ω is denoted by BV(Ω) and it is a Banach space with respect to the norm u BV(Ω) := u L 1 (Ω) + |Du|(Ω).Definition 2.2.Let E be a L n -measurable set.The perimeter of E inside Ω is defined as Per(E, Ω) := |Dχ E |(Ω), and we say that E is a set of finite perimeter in Ω if χ E ∈ BV(Ω).If Ω = R n , we denote Per(E) := Per(E, R n ).
It is also worth mentioning the isoperimetric inequality for sets of finite perimeter.
Theorem 2.1 (Isoperimetric inequality).Let E ⊂ R n be a bounded set of finite measure.Then it holds where ω 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 Clearly it holds for every Borel set A ⊆ Ω.Let us recall the following Fleming-Rishel formula (see [FR60] or [EG15]): Theorem 2.2 (Fleming-Rishel formula).Let Ω ⊂ R n be an open set and let u ∈ BV(Ω), then for almost every t ∈ (−∞, +∞) the set {u > t} has finite perimeter in Ω and it holds (2.1) then u ∈ BV(Ω) and consequently (2.1) holds.

Rearrangements of functions
We now briefly recall some notions about rearrangements.We refer for instance to [Tal94; Kes06; Kaw85] for all the details.
Definition 2.3.Let Ω be a measurable set and let u : Ω → R be a measurable function, the distribution function of u is defined as where, here and throughout the paper, |E| denotes the n-dimensional Lebesgue measure of a measurable set E.
It can be proved that Definition 2.4.Let u : Ω → 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 * and u * are lower semi-continuous; • 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 * (µ(t)) ≤ t for every non-negative t, µ(u * (s)) ≤ s for every non-negative s; • u * (µ(t) − ) ≥ t for every non-negative t, µ(u * (s) − ) ≥ s for every non-negative s; • the Hardy-Littlewood inequality: for any u, v : Definition 2.5.Let u : Ω → R be a measurable function.The Schwarz rearrangement or the spherically symmetric decreasing rearrangement of u is defined as where ω 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 ) is non-negative, radial and radially decreasing (increasing); • u ♯ , u ♯ and u are equally distributed which means they have the same distribution function; • the Polya-Szegö inequality holds true [PS51]: We recall the Theorem of Giarrusso and Nunziante ([GN84, Theorem 2.2]).
Theorem 2.3.Let Ω ⊂ R n be a bounded open set, let Ω ♯ be the centered ball, let p ≥ 1, let f : Ω → R be a measurable function, let H : R n → R be measurable non-negative functions and let K : [0, +∞) → [0, +∞) be a strictly increasing real-valued function such that Let v ∈ W 1,p 0 (Ω) be a function that satisfies denoting by z ∈ W 1,p 0 (Ω ♯ ) the unique spherically decreasing symmetric solution to Moreover, in [Mer97] the following uniqueness result is proved: From now on Ω ⊂ R n is a bounded open set with finite perimeter.Let us consider and u a non-negative function belonging to BV 0 (Ω).Let us define 3) The function f (•, s) belongs to BV 0 (Ω) for every s ∈ [0, +∞) since it is a truncation of u (See [AFP00, Theorem 3.96]).Moreover, for every s ∈ [0, +∞) 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.
Hence, for all 0 ≤ s 1 < s 2 < +∞ we have Per({u > ξ}) dξ where Since this holds for every open interval (s 1 , s 2 ), we have Observing that H is a Lipschitz function, D H(u * ) is given by (see [AD90]) since µ(u * (s)) = s a.e. with respect Du * (by the properties of the rearrangements) and since for s Then we can write dD H(u * ) 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 ⋆ , defined in (1.3), is Now, let us integrate (3.2) between 0 and +∞ 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 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

Two versions of the torsional rigidity
For a given Λ > 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 ψ to |ψ| it holds F (ψ) ≥ F (|ψ|).
Assuming that problem (4.2) admits a minimum u ∈ H 1 0 (Ω), then it is also a maximum for the torsional rigidity defined by Diaz, Polya and Weinstein in [DW48; PW50] of a multiplyconnected cross-section with fixed measure of the holes, that is where A i are the connected component of {|∇u| = 0} and D = Ω ∪ 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 [AC81; DCL89].
However, in the literature, penalizing terms of the form |{|∇ψ| = 0}| 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 Ω is a ball.
Proposition 4.1.Let Λ, R > 0 and let B R be the centered ball with radius R. Then the functional F Λ defined in (4.1) admits a minimizer v belonging to H 1 0 (Ω).Such a minimizer is unique up to a sign, it is radially symmetric and |∇v| is radially increasing.
Proof.We divide the proof in 3 steps.

Boundness from below.
First of all, let us prove that the functional F Λ is bounded from below for every choice of Λ and for every R > 0. For all ψ ∈ H 1 0 (B R ), sing Young and Poincaré inequalities, we get Chosing ε sufficiently small such that 2. Compactness and semicontinuity.Now we consider a minimizing sequence {ψ k } for T F (B R , Λ) and we prove that it is bounded in H 1 0 (B R ).We can assume that F Λ (ψ k ) ≤ −T F (B R , Λ) + 1 and by Proposition 2.3 we can assume that ψ k are radial function with |∇ψ k | radially symmetric increasing.
Using Young and Poincaré inequalities, we obtain we have then by Poincaré inequality, the sequence {ψ k } is bounded in H 1 0 (B R ).This implies that there exists a subsequence (still denoted by ψ k ) and a function v ∈ H 1 0 (B R ) such that ψ k → v strongly in L 2 (Ω), a.e. in Ω and ∇ψ k ⇀ ∇v weakly in L 2 .Let us show that v is a minimum for F Λ .
The lower semicontinuity of the norms gives Let us deal with the last term of F Λ and let us prove that lim inf Denoting by r k the radius of the ball where |∇ψ k | = 0, we can assume that r k converges to some r ≥ 0. Therefore So we have just to prove that |∇v| = 0 in B r .Since {ψ 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 ⊂ B r with |A| > 0 and that |∇v| = 0 in A. Clearly there exists ε > 0 such that Since this must be true for every and since v is minimum, it holds Since |∇v| is equally distributed with |∇v|, the previous equality implies Remark 4.1.We highlight that Theorem 2.3 ensures us that the minimum when Ω is a ball has gradient equal to zero only in a ball B r centered at the origin with 0 ≤ r ≤ R. Now, as already mention in the introduction, we prove a Saint-Venant type inequality for T F (Ω, Λ). 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 G gives the long-time distribution of temperature of the domain Ω and the displacement around Ω of a thin layer of insulator with total mass equal to m.We refer to [BBN17] for more details.
If Ω is a Lipschitz domain, G(ψ) achieves its minimum among all H 1 (Ω) functions.So we define 1 T G (Ω, m) := min It is easy to check that the Euler-Lagrange equation of this functional is Proof.For every function ψ ∈ H 1 (Ω), by 1.2, there exists ψ ∈ H 1 (Ω ♯ ) that satisfies and then T G (Ω, m) ≤ T G (Ω ♯ , m).

Competing Interests
The authors have no competing interests as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.
Funding Information 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.
Author contribution All authors equally contributed to the paper.