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 $${\int }_{\mathrm{\Omega }}(⟨A(x,u)\mathrm{\nabla }u,\mathrm{\nabla }u⟩+\mathrm{\Lambda }(x,u))dx$$ here $A(x,u)=A_{+}(x){\chi }_{\{u>0\}}+A_{-}(x){\chi }_{\{u\le 0\}}$ for some elliptic and bounded matrices $A_{\pm }$ with H"older continuous entries and $\mathrm{\Lambda }(x,u)={\lambda }_{+}(x){\chi }_{\{u>0\}}+{\lambda }_{-}(x){\chi }_{\{u\le 0\}}$. We prove that the free boundaries of minimizers of the above functional touches the fixed boundary $\mathrm{\partial }\mathrm{\Omega }$ in a tangential fashion, provide the graph of boundary data touches its zeros smoothly. This assumption is reflected in the \eqref{DPT} condition.