Calculus of Variations and Geometric Measure Theory

L. Ferreri - L. Spolaor - B. Velichkov

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

created by ferreri on 10 May 2024



Inserted: 10 may 2024

Year: 2024

ArXiv: 2405.05799 PDF


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.