Characterization of balls through optimal concavity for potential functions

Let $p\in(1,n)$. If $\Omega$ is a convex domain in $\rn$ whose $p$-capacitary potential function $u$ is $(1-p)/(n-p)$-concave (i.e. $u^{(1-p)/(n-p)}$ is convex), then $\Omega$ is a ball.


Introduction
Let n ≥ 3, Ω ⊂ R n and p ∈ (1, n). The p-capacity of Ω can be defined as follows (see for instance [8], §4.7): where C ∞ c (R n ) denotes the set of functions from C ∞ (R n ) having compact support. In the sequel Ω is a bounded open convex set, then the above infimum is in fact a minimum which is realized by the (classical) solution u of the following problem The function u is called p-capacitary potential function of Ω and it holds (1.3) Cap p (Ω) = R n \Ω |∇u| p dx .
It is well known that if Ω is (bounded, open and) convex, then u is quasi-concave, that is all its superlevel sets Ω(t) = {x ∈ R n : u(x) ≥ t} t ∈ (0, 1] are convex, see [9,13,11]. In fact, if Ω is smooth and strictly convex, one could even expect u to satisfy some stronger concavity property, namely that there exist some suitable α(Ω) < 0 such that u is α(Ω)-concave, see Section 2.5. We recall here that a positive function is said α-concave, for α < 0, if u α is convex (see again Section 2.5 for more details).
Indeed, when Ω is a ball of radius R > 0 centered at x 0 , it is easy to find explicitly the solution of (1.2), that is and it results to be (−1/q)-concave. In this short note I prove that nothing better is possible and that this power concavity is optimal, in the sense that the property of u −1/q to be convex characterizes balls. Precisely, the main result of this paper is the following.
To prove this theorem we will use three main ingredients: -the first one is the Brunn-Minkowski inequality for p-capacity and its equality condition, proved in [3,5] for p = 2 and in [7] for a generic p; -the second ingredient is an easy relation existing between the p-capacity of a generic level set of u and the capacity of Ω, see formula (2.3); -the third ingredient is the expression of p-capacity through the behaviour at infinity of the potential function, see formula (2.5).
In fact the last ingredient is needed to prove the following property, which may have its own interest and it is new, to my knowledge.
Theorem 1.2. If the solution u of (1.2) has two homothetic level sets, then Ω is a ball.
In particular: if u has a level set that is homothetic to Ω, then Ω is a ball. We recall here that two sets A, B ⊂ R N are said homothetic if there exist ρ > 0 and ξ ∈ R N such that B = ρA + ξ, i.e. if they are dilate and translate of each other.
To some extent, both the problems considered in Theorem 1.1 and Theorem 1.2 fall in the framework of overdetermined problems: in the first case the overdetermination is given by the concavity property of the solution u of (1.2), in the latter case the overdetermination is given by the existence of two homothetic level sets of u.
The paper is organized as follows. Firstly in Section 2 I introduce notation and recall some needed results and formulas (in particular the three main ingredients recalled above). I prove Theorem 1.2 in Section 3. Finally Section 4 contains the proof of Theorem 1.1.

Preliminaries
2.1. Basic notation. If a, b ∈ R N , we denote by a, b their scalar product and by |a| the euclidean norm of the vector a, i.e. |a| = a, a . If M is an n × n symmetric matrix, we denote by tr(M ) and det(M ) its trace and its determinant respectively; M > 0 means that M is positive definite, M T is the transposed of M and M −1 its inverse.
If C is a subset of R n , |C| is its Lebesgue measure, C is its closure, int(C) is its interior and ∂C is its boundary For r > 0 and x ∈ R N , we denote by B(x, r) the ball of radius r centered at x. Then we set If u is a twice differentiable function, by Du and D 2 u we denote, as usual, the gradient of u and its Hessian matrix respectively, i.e. Du = ( ∂u ∂x 1 , . . . , ∂u ∂x N ) and D 2 u = ( ∂ 2 u ∂x i ∂x j ) N i,j=1 and we denote by ||u|| L p (Ω) the L p norm of the function u : Ω → R.
2.2. Ingredient 1: the Brunn-Minkowski inequality for p-capacity. The original form of the Brunn-Minkowski inequality involves volumes of convex bodies (i.e. compact convex subsets of R n with non-empty interior) and states that Vol n (·) 1/n is a concave function with respect to the Minkowski addition, i.e.
1 n for every convex bodies K 1 and K 2 and λ ∈ [0, 1]. Here Vol n is the n-dimensional Lebesgue measure and the Minkowski addition of convex sets is defined as follows while λA = {λx : x ∈ A} for any λ ∈ R, as usual. Inequality (2.1) is one of the fundamental results in the modern theory of convex bodies; it can be extended to measurable sets and several other important inequalities, e.g. the isoperimetric inequality, can be deduced from it.
Suitable versions of the Brunn-Minkowski inequality hold also for the other quermassintegrals (see [18,10]) and recently Brunn-Minkowski type inequalities have been proved for several important geometric and analytic functionals (see for instance [3,4,6,7,14,16,17] and especially the beautiful survey paper [10]). Notice that in all the known cases, equality conditions are the same as in the classical Brunn-Minkowski inequality for the volume, i.e. equality holds if and only if the involved sets are (convex and) homothetic (i.e. translate and dilate of each other).
We will use the following theorem from [7].
Roughly speaking (2.2) says that Cap p (·) 1 n−p is a concave function in the class of convex bodies endowed with the Minkowsky addition. But what is most relevant to the present paper is the equality condition: if equality holds in (2.2), then K 1 and K 2 are homothetic.
We recall that in the case of the Newton capacity, i.e. for p = 2 and n ≥ 3, inequality (2.2) was proved by C. Borell [3] and more recently in [5] L.A. Caffarelli, D. Jerison and E.H. Lieb treated the equality case. In [7] the treatments of the inequality and of its equality case are unified and the results are extended to a generic p ∈ (1, n). An analogous relation holds in the generic case p ∈ (1, n) refer to [7] for instance.

2.5.
Power-concave and quasi-concave functions. For α = 0 a function v : Furthermore, v is said quasi-concave if all its super level sets {x ∈ R N : v(x) ≥ t} are convex. To some extent, quasi-concavity corresponds to α-concavity when α = −∞.
It is easily seen that if v is α-concave (for some α ∈ R), then v is β-concave for every β ≤ α. Moreover, it is obvious that every α-concave function (for some α ∈ R) is quasi-concave.
Given a quasi-concave function v, it is then natural to ask if it satisfies some better concavity properties and following [12] we define the concavity number of u as follows By [12,Property 5], for any C 2 quasi-concave function it is possible to explicitly calculate where v θ and v θθ denote respectively the first and the second derivatives of v in direction θ.
According to [1,7,15] it is also possible and useful to associate to any quasi-concave function v a support function h v : R × R N → R, such that h v (X, t) is the support function of the super level set {v ≥ t} calculated at X, i.e.
In this way the concavity of v corresponds to the concavity of h v with respect to t, that is v results to be concave if and only if Consequently v is α-concave (for some α = 0) if and only if since v is quasi-concave. Then we can also write As already said in the Introduction, it is well known that when Ω is convex, its p-capacitary potential u is quasi-concave; then we set α(Ω) = α(u) .
If Ω is sufficiently regular and strictly convex, one can expect that α(Ω) > −∞. The aim of this paper is to prove the following: if α(Ω) = (1 − p)/(n − p), then Ω is necessarily a ball.
and from (3.4) we easily obtain Then u is radial in R N \ Ω(s) and, by analytic continuation, it is radial in R N \ Ω and Ω is a ball.
We will proceed by proving that, if u is q-concave, then all its level sets are homothetic. Then the proof will be concluded thanks to Theorem 1.2.