The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem

Abstract

We consider the p-Laplacian equation −∆pu = 1 for 1 < p < 2, on a regular bounded domain Ω ⊂ R N , with N ≥ 2, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature H of ∂Ω is constant, then Ω is a ball and the unique solution of the Dirichlet p-Laplacian problem is radial. The main tools used are integral identities, the P -function, and the maximum principle.

Some evaluations of the fractional $p$-Laplace operator on radial functions

“…which proves the thesis also in this case. Having in mind that, for p = 2, the fractional Laplacian of u s,2 (x) = (1 − |x| 2 ) s + is constant in (−1, 1), see for instance Section 3, and that −∆ p (1 − |x| p p−1 ) is constant in (−1, 1), see for instance [6], it is tempting to conjecture that also (−∆ p ) s u s,p is constant in (−1, 1). In the next section we verify numerically that this conjecture is false.…”

“…On the other hand, in the local case s = 1, it is easy to prove that the function (1 − |x| m ), with m = p p−1 , has constant p-Laplacian in (−1, 1), see for instance [6]. Moreover, in the linear case p = 2, it has been proved that (1 − x 2 ) s + satisfies (−∆) s (1 − x 2 ) s + = Const.…”