The obstacle problem with singular coefficients near Dirichlet data

Abstract

In this paper we study the behaviour of the free boundary close to its contact points with the fixed boundary $B\cap \{x_1=0\}$ in the obstacle type problem

where $a<1$, $B^{+}=B\cap \{x_1>0\}$, B is the unit ball in ${\mathbb{R}}^n$ and $n\ge 2$ is an integer.

Let $\mathrm{\Gamma }=B^{+}\cap \partial \{u>0\}$ be the free boundary and assume that the origin is a contact point, i.e. $0\in \overline{\mathrm{\Gamma }}$. We prove that the free boundary touches the fixed boundary uniformly tangentially at the origin, near to the origin it is the graph of a $C^1$ function and there is a uniform modulus of continuity for the derivatives of this function.