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

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.