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 $\Delta u = \lambda_{+\chi_{\{ u>0 \}}} - \lambda_{-\chi_{\{ u<0 \}}},\quad\lambda_{\pm}>0. $ We prove that the free boundary touches the fixed boundary (uniformly) tangentially if the boundary data f and its first and second derivatives vanish at the touch-point.