The two membranes problem for different operators

TL;DR

This paper investigates the regularity and free boundary properties of solutions to the two membranes problem involving different, possibly nonlocal, operators, introducing a viscosity approach and establishing optimal regularity results.

Contribution

It develops a viscosity solution framework for the two membranes problem with different operators and proves new regularity and free boundary characterizations.

Findings

Solutions are Hölder continuous.

Solutions are $C^{1,eta}$ regular when operators have different orders.

Optimal regularity and free boundary characterization when one operator is fractional Laplacian.

Abstract

We study the two membranes problem for different operators, possibly nonlocal. We prove a general result about the H\"older continuity of the solutions and we develop a viscosity solution approach to this problem. Then we obtain $C^{1,\gamma}$ regularity of the solutions provided that the orders of the two operators are different. In the special case when one operator coincides with the fractional Laplacian, we obtain the optimal regularity and a characterization of the free boundary.