Bernoulli Free Boundary Problem for the Infinity Laplacian

Abstract

We study the interior Bernoulli free boundary problem for the infinity Laplacian. Our results cover existence, uniqueness, and characterization of solutions (above a threshold representing the "infinity Bernoulli constant"), their regularity, and their relationship with the solutions to the interior Bernoulli problem for the p-Laplacian.

On the supremal version of the Alt–Caffarelli minimization problem

“…Before giving the proofs of Propositions 9 and 10, we need to recall a result from our paper [9] about the Bernoulli problem (4) on convex domains (therein, also the more general case of non-convex domains is considered). We first resume a few preliminary definitions.…”

“…Indeed, if (P ) Λ admits an infinity harmonic solution, it agrees necessarily with w r Λ because its positivity set is uniquely determined by (9). To see that w r Λ is actually a solution of (P ) Λ , we observe that it has the same positivity set as v r Λ , and a Lipschitz constant not larger than v r Λ (because w r Λ has the AML property mentioned in the Introduction).…”

“…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 ∞ .…”

“…For a more precise description of the applied side, including several references, see [13,Section 2]. In our recent paper [9], we have studied the existence and uniqueness of solutions to the following 'Bernoulli problem for the ∞-laplacian'…”

“…This formula is obtained with the help of the Brunn-Minkowski inequality, and allows to compute explicitly the value of Λ Ω,∞ , at least for simple geometries. In particular, if we compare Λ Ω,∞ with the '∞-Bernoulli constant', defined in [9] by λ Ω,∞ := inf{λ > 0 : (4) admits a non-constant solution} , it turns out that Λ Ω,∞ ≥ λ Ω,∞ ; the computation on balls reveals that the inequality can be strict. Still using its geometric characterization, we prove that Λ Ω,∞ satisfies an isoperimetric inequality, namely that it is minimal on balls under a volume constraint (see Theorem 6).…”