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

Abstract

In this article we study functionals of the following type $${\int }_{\Omega }(⟨A(x,u)\nabla u,\nabla u⟩+\Lambda (x,u))dx$$ here $A(x,u)=A_{+}(x){\chi }_{\{u>0\}}+A_{-}(x){\chi }_{\{u\le 0\}}$ for some elliptic and bounded matrices $A_{\pm }$ with Hölder 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.