The two membranes problem for fully nonlinear operators

TL;DR

This paper investigates the two membranes problem involving fully nonlinear operators, establishing existence, regularity results, and illustrating limitations in free boundary regularity.

Contribution

It introduces a viscosity formulation for the problem, proves existence of solutions, and demonstrates optimal regularity results for Pucci extremal operators.

Findings

Existence of solutions for the two membranes problem with fully nonlinear operators.

Proved optimal $C^{1,1}$ regularity for Pucci extremal operators.

Provided an example showing no general free boundary regularity.

Abstract

We study the two membranes problem for two different fully nonlinear operators. We give a viscosity formulation for the problem and prove existence of solutions. Then we prove a general regularity result and the optimal $C^{1,1}$ regularity when the operators are the Pucci extremal operators. We also give an example that shows that no regularity for the free boundary is to be expected to hold in general.