On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem

Abstract

In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ball $Δu = λ_{+}χ_{\{u>0\}}-λ_{-}χ_{\{u<0\}}, λ_\pm>0$. We prove that the free boundary touches the fixed one in (uniformly) tangential fashion if the boundary data $f$ and its first and second derivatives vanish at the touch-point.