On the fine structure of the solutions to nonlinear thin two-membrane problems in 2D

TL;DR

This paper establishes a detailed structural understanding of solutions to nonlinear thin two-membrane problems in 2D, revealing finite branching points and linking solutions to linear obstacle problems using quasi-conformal maps.

Contribution

It introduces a novel approach connecting nonlinear membrane solutions to linear obstacle problems via quasi-conformal maps, and proves finiteness of branching points.

Findings

Branching points are finite in number.

Difference of sheets is topologically equivalent to linear obstacle problem solutions.

Methods apply to free boundary problems near analytic boundaries.

Abstract

We prove a structure theorem for the solutions of nonlinear thin two-membrane problems in dimension two. Using the theory of quasi-conformal maps, we show that the difference of the sheets is topologically equivalent to a solution of the linear thin obstacle problem, thus inheriting its free boundary structure. More precisely, we show that even in the nonlinear case the branching points can only occur in finite number. We apply our methods to one-phase free boundaries approaching a fixed analytic boundary and to the solutions of a one-sided two-phase Bernoulli problem.