The two membranes problem in a regular tree

TL;DR

This paper investigates the two membranes problem on a regular tree using mean value operators, establishing existence of solutions and analyzing the structure of the coincidence set under certain boundary conditions.

Contribution

It introduces a novel approach to the two membranes problem on regular trees using mean value formulas, providing existence results and geometric properties of solutions.

Findings

Existence of solutions under boundary and source conditions

Finite coincidence set when boundary data are strictly separated

Coincidence set is separated from the boundary in certain cases

Abstract

In this paper we study the two membranes problem for operators given in terms of a mean value formula on a regular tree. We show existence of solutions under adequate conditions on the boundary data and the involved source terms. We also show that, when the boundary data are strictly separated, the coincidence set is separated from the boundary and thus it contains only a finite number of nodes.