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 $\Delta u={\lambda }_{+}{\chi }_{\{u>0\}}-{\lambda }_{-}{\chi }_{\{u<0\}},{\lambda }_{\pm }>0$. 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.

14 pages, 3 figures