An isoperimetric inequality related to a Bernoulli problem

Abstract

Given a bounded domain Ω we look at the minimal parameter Λ(Ω) for which a Bernoulli free boundary value problem for the p-Laplacian has a solution minimising an energy functional. We show that amongst all domains of equal volume Λ(Ω) is minimal for the ball. Moreover, we show that the inequality is sharp with essentially only the ball minimising Λ(Ω). This resolves a problem related to a question asked in [Flucher et al.,

On the supremal version of the Alt–Caffarelli minimization problem

“…The variational p-Bernoulli constant Λ p considered by Kawohl and Daners in [10] agrees with the infimum of positive λ such that problem (14) (with Λ = 1) admits a non-constant solution. In the limit as p → +∞, Λ p does not converge to Λ ∞ (in fact, in [9] we proved that lim p→+∞ Λ p = 1/R Ω ), and this is precisely the reason why, as mentioned in Remark 7, Theorem 6 cannot be obtained by passing to the limit as p → +∞ in the isoperimetric inequality proved by Daners-Kawohl in [10]. In view of Theorem 11, one should not be surprised by the missed convergence of Λ p to Λ ∞ .…”

“…To make the difference is the gradient term: the integral functional appearing in (2) is converted into the supremal functional ∇u ∞ in (P ) Λ . Problem (2) has a long history: starting from the groundbreaking paper [1], where it was introduced in the linear case p = 2, it has been widely studied in later works for any p ∈ (1, +∞) (see for instance [6,10,11,15]). In particular, the topic which has been object of a thorough investigation is the regularity of the free boundary, which has been settled to be locally analytic except for a H n−1 -negligible singular set [1,11].…”

Bernoulli Free Boundary Problem for the Infinity Laplacian

“…• for every λ ≥ λ Ω,p , problem (24) admits a classical non-constant solution u ∈ C(Ω + (u))∩ C 2 (Ω + (u)), which has convex level sets; moreover, the free boundary F (u) is of class • As a consequence of the results recalled at the above item, we have that (28) Λ Ω,p ≥ λ Ω,p ; this inequality may be strict, as the explicit computation of both constants Λ Ω,p and λ Ω,p in case of the ball reveals [25,Section 4].…”

Krahn’s proof of the Rayleigh conjecture revisited

“…Note that Steps 1-4 do not depend on looking at an eigenvalue problem. The results apply to Schwarz symmetrisation of functions in general and can be used for other purposes like finding the best constant in Sobolev inequalities as in [40] or for nonlinear problems such as [14].…”