Tangential contact of free boundaries and the fixed boundary for variational solutions to a free transmission Problem

TL;DR

This paper investigates the behavior of free boundaries in a variational problem involving transmission conditions, showing they touch the fixed boundary tangentially and that boundary data interacts smoothly with zeros.

Contribution

It establishes the tangential contact of free boundaries with fixed boundaries for a class of variational problems with transmission conditions, under specific regularity assumptions.

Findings

Free boundaries touch the fixed boundary tangentially.

Boundary data graph touches zeros smoothly.

Provides conditions ensuring tangential contact.

Abstract

In this article we study functionals of the following type $$ \int_{\Omega} \Big ( \langle A(x,u)\nabla u, \nabla u\rangle + \Lambda (x,u) \Big )\,dx $$ here $A(x,u)= A_+(x)\chi_{\{u>0\}}+A_-(x) \chi _{\{u\leq 0\}}$ for some elliptic and bounded matrices $A_{\pm}$ with H\"older continuous entries and $\Lambda(x,u) = \lambda_+(x) \chi_{\{u>0\}} + \lambda_-(x) \chi_{\{u\le 0\}}$. We prove that the free boundaries of minimizers of the above functional touches the fixed boundary $\partial \Omega$ in a tangential fashion, provide the graph of boundary data touches its zeros smoothly. This assumption is reflected in the \eqref{DPT} condition.