Free boundary regularity for a degenerate problem with right hand side

Abstract

We consider an one-phase free boundary problem for a degenerate fully non-linear elliptic operators with non-zero right hand side. We use the approach present in [DeS] to prove that flat free boundaries and Lipschitz free boundaries are C 1,γ . keywords: free boundary problems, degenerate fully non-linear elliptic operators, regularity theory.

Recent Results on Nonlinear Elliptic Free Boundary Problems

“…We are interested in the regularity of the free boundary for viscosity solutions of (10), following the strategy introduced in [22]. The same technique was applied to the p-Laplace operator (p(x) ≡ p in (10)), with p ≥ 2, in [56].…”

Recent Results On Nonlinear Elliptic Free Boundary Problems

“…We are interested in the regularity of the free boundary for viscosity solutions of (18), following the strategy introduced in [22]. The same technique was applied to the p-Laplace operator (p(x) ≡ p in ( 18)), with p ≥ 2, in [56].…”

Bernoulli Free Boundary Problem for the Infinity Laplacian

“…Thus, even at the boundary, a viscosity interpretation seems to be the most convenient one in order to manage both existence and uniqueness questions. More precisely, throughout the paper we interpret solutions to (P ) λ to mean viscosity solutions defined in the next definition, introduced by De Silva [28, Definitions 2.2 and 2.3], and has been adopted in several subsequent works (see for instance [29,45,46]). If u, v : Ω → R are two functions and x ∈ Ω, by u ≺ x v we mean that u(x) = v(x) and u(y) ≤ v(y) in a neighborhood of x.…”