Rigidity results for the $p$-Laplacian Poisson problem with Robin boundary conditions

Let $\Omega \subset \mathbb{R}^n$ be an open, bounded and Lipschitz set. We consider the Poisson problem for the $p-$Laplace operator associated to $\Omega$ with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison stated in \cite{AGM}. We prove that the equality is achieved only if $\Omega$ is a ball and both the function $u$ and the right hand side $f$ of the Poisson equation are radial.


Introduction
Symmetrization techniques in the context of qualitative properties of solutions to second-order elliptic boundary value problems are introduced by Talenti in [25].In this seminal paper, the author considers an open, bounded and Lipschitz set Ω ⊂ R n , the ball Ω ♯ with the same measure as Ω and the solutions u and v to the following problems where f ∈ L 2 (Ω) is a positive function and f ♯ is its Schwarz rearrangement of f (see Definition 2.3).In this setting, Talenti proves the following point-wise estimate: For sake of completeness, we observe that this result is proved more generally for a uniformly elliptic linear operator in divergence form.

2
A version of this result for nonlinear operators in divergence form is contained in [26], which includes as a special instance the case of the p-Laplace operator.Moreover, these results are later extended, for instance, to the anisotropic elliptic operators in [2], to the parabolic case in [4], and to higher order operators in [9,28].
Once a comparison result holds, it is natural to ask whether the equality cases can be characterized and, so, if a rigidity result is in force.In [3], the rigidity result linked to problem (1) is proved.Indeed, the authors prove that if equality holds in (2), then Ω is a ball, u is radially symmetric and decreasing and f = f ♯ .Rigidity results for a generic linear, elliptic second order operator can be found in [17] and [18].To the best of our knowledge, rigidity results for nonlinear operators with Dirichlet boundary conditions are not present in the literature.In this paper, we obtain, as a corollary of our results, the rigidity for the p−Laplace operator with Dirichlet boundary conditions in any dimension (see Corollary 3.3).
For a long time, it was believed that comparison results could not be proved by means of spherical rearrangement argument when dealing with Robin boundary conditions, until the recent paper [5].The authors consider the following problems and they prove a comparison result involving Lorentz norms of u and v whenever f is a non negative function in L 2 (Ω) and β is a positive parameter.In particular, in the case f ≡ 1, they prove and, if n = 2, the pointwise comparison holds: In [21], it is proved that (3) is rigid, i.e. the equality case is possible if and only if Ω is a ball and u is a radial and decreasing function.
Generalizations of the results contained in [5] can be found for the anisotropic case in [24], for mixed boundary conditions in [1], in the case of the Hermite operator in [13].
In the present paper, we focus our study on the rigidity for p−Laplace operator.In this case, the comparison results are obtained in [6] and the setting is the following.
Let Ω be a bounded, open and Lipschitz set in R n , with n ≥ 2. Let p ∈ (1, +∞) and let f ∈ L p ′ (Ω) be a positive function, where p ′ = p/(p − 1).The Poisson problem for the p−Laplace operator with Robin boundary conditions is where ν is the unit exterior normal to ∂Ω and β > 0. A function u ∈ W 1,p (Ω) is a weak solution to (4) if The symmetrized problem associated to (4) is the following In [6] the authors establish a comparison result between suitable Lorentz norms (see Definition 2.4) of the solutions u and v to problems (4) and ( 6) respectively.In particular, they prove and in the case f ≡ 1, they prove and In the present paper, we want to characterize the equality case in ( 7) and ( 9), answering to the open problem contained in [21].For simplicity, we state the main Theorem only in the case f ∈ L p ′ (Ω) positive, since in the case f ≡ 1 the proof is analogous, as we observe in Remark 4.1.
Theorem 1.1.Let Ω ⊂ R n be a bounded, open and Lipschitz set and let Ω ♯ be the ball centered at the origin with the same measure as Ω.Let u be the solution to (4) and let v be a solution to (6).
then, there exists x 0 ∈ R n such that The idea of the proof of Theorem 1.1 is the following.First of all, we prove that hypothesis (10) implies that the superlevel sets of u are balls.The main difficulty is to prove that these balls are concentric.
Differently from the case of the Laplace operator with Dirichlet boundary conditions studied in [4,16], we can't apply directly the steepest descent method introduced in [8], because it strongly relays on the continuity of both the solution and of its gradient.In the case of the p−Laplace equation, the continuity of the solution up to the boundary depends on the regularity of the given datum f .To overcome this regularity issue we show that u is a solution to a suitable Dirichlet problem and it satisfies the Pólya-Szegő inequality with equality sign.Then, we can conclude that u is radially symmetric and decreasing, using the classical result contained in [10].We make use of Lemma 3.2, where the rigidity of the Poisson problem for the p-Laplace operator with Dirichlet boundary condition is proved under the assumption f ∈ L p ′ (Ω) and positive.Up to our knowledge, Corollary 3.3 seems to be new in the literature.
The paper is organized as follows.In Section 2 we recall some definitions about rearrangement of functions and we state some lemmas that we will need in the proof of the main theorem.Section 3 is dedicated to the proof of the main result and we conclude with a list of open problems.

Notation and Preliminaries
Throughout this article we will denote by |Ω| the Lebesgue measure of an open and bounded Lipschitz set of R n , with n ≥ 2, and by P (Ω) the perimeter of Ω.Since we are assuming that ∂Ω is Lipschitz, we have that P (Ω) = H n−1 (∂Ω), where H n−1 denotes the (n − 1)−dimensional Hausdorff measure.
We recall the classical isoperimetric inequality and and we refer the reader, for example, to [22,11,12,27] and to the original paper by De Giorgi [15].

Theorem 2.1 (Isoperimetric Inequality)
. Let E ⊂ R n be a set of finite perimeter.Then, where ω n is the measure of the unit ball in R n .Equality occurs if and only if E is (equivalent to) a Ball.
For the following theorem, we refer to [7].
Let us recall some basic notions about rearrangements.For a general overview, see, for instance, [19].We have the following relation between u ♯ and u * : where ω n is the measure of the unit ball in R n , and one can easily check that the functions u, u * e u ♯ are equi-distributed, i.e. they have the same distribution function, and it holds We also recall the Hardy-Littlewood inequality, an important propriety of the decreasing rearrangement, We now introduce the Lorentz spaces (see [28] for more details on this topic).
Definition 2.4.Let 0 < p < +∞ and 0 < q ≤ +∞.The Lorentz space L p,q (Ω) is the space of those functions such that the quantity: Let us observe that for p = q the Lorentz space coincides with the L p space, as a consequence of the Cavalieri's Principle The solutions u to problem (4) and v to problem (6) are both p-superharmonic and, as a consequence of the strong maximum principle and the lower semicontinuity (see [29,20]), they achieve their minima on the boundary.If we set the positiveness of β and the Robin boundary conditions leads to u m ≥ 0 and v m ≥ 0. Hence, u and v are strictly positive in the interior of Ω.Moreover, we can observe that indeed, Now, for t ≥ 0, we introduce the following notations: Because of the invariance of the p−Laplacian and of the Schwarz rearrangement of f by rotation, the solution v to ( 6) is radial, so the set V t are balls.Now, we recall some technical Lemmas, proved in [6], that we need in what follows.We recall the proof of Lemma 2.3 for reader's convenience, while we omit the proof of Lemma 2.4 and Lemma 2.5.

Lemma 2.3.
Let u be the solution to (4) and let v be the solution to (6).Then, for almost every t > 0, we have and where Proof.Let t > 0 and h > 0. In the weak formulation (5), we choose the following test function obtaining Dividing ( 18) by h, using coarea formula and letting h go to 0, we have that for a.e.t > 0 Using the isoperimetric inequality, for a.e.t ∈ [0, +∞) we have and, so, (15) follows.Finally, we notice that, if v is the solution to ( 6), then all the inequalities above are equalities, and, consequently, we have (16).
Lemma 2.4.For all τ ≥ v m , we have Moreover, Lemma 2.5 (Gronwall).Let ξ(τ ) be a continuously differentiable function, let q > 1 and let C be a non negative constant C such that the following differential inequality holds Then, we have and The following Lemma is contained in [4].
then, for every τ ≥ 0 there exists t ≥ 0 such that we have, up to zero measure set, We conclude this preliminary session, recalling the classical results contained in [10] (see Theorem 1.1 and Lemma 2.3).In particular, the result contained in (iii) of Lemma (2.7) gives the rigidity of the Pólya-Szegő inequality (see [23]): Theorem 2.7.Let w ∈ W 1,p (R n ), let σ(t) be its distribution function and let Then, the following are true: i.For almost all t ∈ (0, w M ), iii.If and (32) holds, then there exist a translate of w ♯ which is almost everywhere equal to w.
Remark 2.2.We observe that in [14], it is proved that the condition implies (32).So, by (iii) in Lemma 2.7, if we have (33) and (34), there exists a translated of w ♯ which is almost everywhere equal to w.

Proof of Theorem 1.1
In order to prove the main Theorem 1.1, we divide the proof into the following steps.First of all, we prove that, under the assumptions of Theorem 1.1, equality holds in (15) and this is the content of Proposition 3.1.Then, in Proposition 3.4, we prove that equality in (15) implies the fact that Ω is a ball and u and f are radial functions.In order to prove this last step, we need the key Lemma 3.2.
Proposition 3.1.Let u be the solution to (4) and let v be the solution to (6).
then equality holds in (15) for almost every t.
Proof.Since we are assuming that u L pk,p (Ω) = v L pk,p (Ω ♯ ) , we have that Let us multiply (15) by , and let us integrate from 0 to +∞: where in the last inequality we have used µ(t) ≤ |Ω| and (24) in Lemma 2.4.As far as v is concerned, it holds We observe that the left-hand-side of (36) and the left-hand-side of (37) are equal from (35).So, it follows Setting dω, and integrating (38) by parts, we get In [6] (see the proof of Theorem 1.1), it is proved that and we recall here the proof for the reader's convenience.In order to do that, we multiply (15) by t p−1 F (µ(t))µ(t) 1) and we integrate between 0 and τ > v m .First, we observe that, by 1) is non decreasing.
Hence, we obtain If we integrate by parts both sides of the last expression and we set where Setting now So, Lemma 2.5, with τ 0 = v m and q=2, gives Of course, the inequality holds as equality if we replace µ(t) with φ(t), so we get: So, we get equality in (36) and, consequently, in (15) for almost every t, indeed In the following Lemma we prove that a solution to a Dirichlet problem, such that its distribution function satisfies the differential equation ( 42), is necessarily defined on a ball and it has to be radial and decreasing.Lemma 3.2.Let Ω ⊂ R n be an open, bounded and Lipschitz set.Let f ∈ L p ′ (Ω) be a positive function, let w be a weak solution to and let σ be the distribution function of w.If σ satisfies the following condition then, there exists x 0 such that Proof.First of all, we recall that w is a weak solution to (41) if and only if Arguing as in the proof of (15) in Lemma 2.3, choosing the same test function ϕ, defined in (17), where W t = {x ∈ Ω : w(x) > t}.
If we apply the isoperimetric inequality to the superlevel set W t , the Hölder inequality and the Hardy-Littlewood inequality, we get, for almost every t, nω So, hypothesis (42) ensures us that equality holds in the isoperimetric inequality (45), in the Hölder inequality (46) and in the Hardy-Littlewood inequality (47).We now divide the proof into three steps.
Step Since { w > t } = ∪ k { w > t k } can be written as an increasing union of balls, {w > t} is a ball for all t and, in particular, Ω = {w > 0} is a ball too and we obtain that Ω = x 0 + Ω ♯ .
From now on, we can assume without loss of generality that x 0 = 0.
Step 2. Let us prove that the superlevel sets are concentric balls.Equality in (46) implies also equality in Hölder inequality, i.e.
This means that, for almost every t, |∇w| is constant H n−1 −almost everywhere on ∂W t , and we denote by C t the (H n−1 −a.e.) constant value of |∇w| on ∂W t .We claim that C t = 0 for almost every t.Indeed, (44) and the positivity of f ensure us that Integrating (42), we obtain w ♯ (x) = z(x), for all x ∈ Ω ♯ , where z is the solution to and it has the following explicit form: Using (ii) in Lemma 2.7, we have that (49) implies the absolutely continuity of σ.Now, we denote by C ♯ t the (H n−1 −a.e.) constant value of ∇w ♯ on ∂W ♯ t .We recall that it holds and, by the absolutely continuity of σ, we have Since w and w ♯ are equi-distributed, we have, So, by the coarea formula, we get By (iii) in Lemma 2.7, we conclude that u = u ♯ .
Step 3. Let us prove that f is radial and decreasing.Equality in (47) reads, for almost every t, Moreover, for all τ ∈ [0, w M ), there exists a sequence { τ k } such that and, by the continuity of σ(•), we have ˆσ(τ) By Lemma 2.6, we have that for all τ , there exists α τ such that Consequently, we have that also f is radial and decreasing, so f = f ♯ .
As a direct consequence of Lemma 3.2, we obtain the rigidity for the p−Laplace operator with Dirichlet boundary conditions.
Proof.From the proof of Lemma 3.2, it follows that the distribution function of w, denoted by σ, satisfies Now, we integrate (51) from 0 to t, obtaining So, if w ♯ = z, we have w * = z * , and consequently we obtain equality in (51) for almost every t ∈ [0, w M ].We can conclude by applying Lemma 3.2.Now, using Lemma 3.2, we are in position to conclude the proof of the main Theorem.Proposition 3.4.Let Ω ⊂ R n be an open, bounded and Lipschitz set and let Ω ♯ be the ball with the same measure as Ω.Let u be the solution to (4) and let µ be its distribution function.If equality holds in (15), then there exists x 0 ∈ R n such that Proof.Firstly, we claim that the superlevel sets { u > t } are balls for every t ∈ [0, u M ).Equality in (15) implies the equality in (20) So, using the fact that we have equality in (15) by hypothesis, we get We conclude now with the proof of the main Theorem.
Proof of Theorem 1.1.From Proposition 3.1, we have that the hypothesis of Theorem 1.1 u L pk,p (Ω) = v L pk,p (Ω ♯ ) , for some k ∈ ò 0, n(p − 1) (n − 2)p + n ò implies the following equality for almost every t ∈ (0, u M ) where µ(t) is the distribution function of u.Now, we are in position to apply Proposition 3.4, and, so, there exists x 0 ∈ R n such that

Remarks and open problems
Remark 4.1.In [6] the authors also prove that in the case f ≡ 1, it holds We stress that the proof of Theorem 1.1 can be adapted to case f ≡ 1, regardless of the fact that now the admissible k varies in a wider range.
Open problem 4.2.Below we present a list of open problems and work in progress.
• Generalize the rigidity results in the anisotropic setting, starting from the comparison proved in [24].
• Generalize the rigidity results to other problems, such as the ones investigated in [1], [13].

Definition 2 . 1 .Definition 2 . 2 .Remark 2 . 1 .Definition 2 . 3 .
Let u : Ω → R be a measurable function, the distribution function of u is the function µ : [0, +∞[ → [0, +∞[ defined as the measure of the superlevel sets of u, i.e.µ(t) = |{ x ∈ Ω : |u(x)| > t }|.Let u : Ω → R be a measurable function, the decreasing rearrangement of u is the distribution function of µ.We will denote it by u * (•).Let us notice that the function µ(•) is decreasing and right continuous and the function u * (•) is its generalized inverse.The Schwartz rearrangement of u is the function u ♯ whose superlevel sets are balls with the same measure as the superlevel sets of u.

Corollary 3 . 3 .
Let Ω ⊂ R n be an open, bounded and Lipschitz set.Let f ∈ L p ′ (Ω) be a positive function and let w and z be weak solutions respectively to
1.Let us prove that the superlevel set { w > t } is a ball for all t ∈ [0, w M ).Equality in (45) implies that, for almost every t, W t is a ball.On the other hand, for all t ∈ [0, w M ), there exists a sequence { t k } such that 1. t k → t; k } is a ball for all k.
, i.e. = P (U t ), for a. e. t ∈ [0, u M ] that means that almost every superlevel set is a ball.Arguing as in Step 1 of Lemma 3.2, we can conclude that every superlevel set is a ball, so, Ω = {u > u m } is a ball and we obtain that Ω = x 0 + Ω ♯ .Let us observe that for every t, s ∈ [u m , u M ] with t < s, as both U t and U s are balls, we have that ∂U t ∩ ∂U s contains at most one point.In particular, the function w = u − u m is a weak solution to the Dirichlet problem (41) in Ω.We claim that σ(t) = |{ w > t }| satisfies (42).Since { w > t } = { u > t + u m }, we have σ(t) = µ(t + u m ) for all t ∈ [0, u M − u m ].Moreover, we have