An overdetermined problem for the anisotropic capacity

Abstract

We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in RN, establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of Reichel (Arch Ration Mech Anal 137(4):381–394, 1997), where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm H. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm H0)

Some sphere theorems in linear potential theory

“…Reichel's Theorem has been generalized in a number of directions. A different proof of Theorem 1.5 has appeared in [BCS16], and similar results have been proved for other classes of elliptic inequalities [Rei96], for annulus-type domains [Rei95] and for the p-laplacian [GS99,BC17]. The case of a disconnected Ω -with possibly different Dirichlet and Neumann condition on the different boundary components-is analyzed in [Sir01].…”

A Note on an Overdetermined Problem for the Capacitary Potential

“…This kind of problem has been successfully solved in [9] by using the method of moving planes. Other similar problems and related results can be found in [2,3,7,10,11,12]. In this note we discuss two kinds of overdeterminations involving the mean curvature H ∂Ω of ∂Ω (that is the average of the principal curvatures of ∂Ω).…”

“…Theorem 1.1. Let Ω ⊂ R n be a bounded domain with boundary of class C 2 and let u be the solution of (3). If u and Ω are such that…”

Gradient estimates for the perfect conductivity problem in anisotropic media

“…Moreover, these kinds of anisotropy are of strong interest in elasticity, noise-removal procedures in digital image processing, crystalline mean curvature flows and crystalline fracture theory. The literature is very wide and we just mention [12,13,14,15,16,18,21,22,23,29,34,37,40,41] and references therein for an interested reader.…”