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 $\int_Ω \Big ( \langle A(x,u)\nabla u, \nabla u\rangle + Λ(x,u) \Big )\,dx$ here $A(x,u)= A_+(x)χ_{\{u>0\}}+A_-(x) χ_{\{u\leq 0\}}$ for some elliptic and bounded matrices $A_{\pm}$ with Hölder continuous entries and $Λ(x,u) = λ_+(x) χ_{\{u>0\}} + λ_-(x) χ_{\{u\le 0\}}$. We prove that the free boundaries of minimizers of the above functional touches the fixed boundary $\partial Ω$ in a tangential fashion, provide the graph of boundary data touches its zeros smoothly. This assumption is reflected in the \eqref{DPT} condition.

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