Regularity properties of a free boundary near contact points with the fixed boundary

Abstract

In the upper half of the unit ball $B\sp + =\{ |x|<1,x\sb 1>0\}$, let $u$ and $\Omega$ (a domain in $\mathbf {R}\sp n\sb + =\{x\in \mathbf {R}\sp n : x\sb 1>0\}$) solve the following overdetermined problem:

\Delta u =\chi_\Omega\quad \text{in}\ B^+, \qquad u=|\nabla u |=0 \quad \text{in}\ B^+\setminus \Omega, \qquad u=0 \quad \text{on}\ \Pi\cap B,

where $B$ is the unit ball with center at the origin, $\chi\sb \Omega$ denotes the characteristic function of $\Delta,\Pi=\{ x\sb 1=0\} ,n\geq 2$, and the equation is satisfied in the sense of distributions. We show (among other things) that if the origin is a contact point of the free boundary, that is, if $u(0)=|\nabla u(0)|=0$, then $\partial\Delta\cap B\sb {r\sb 0}$ is the graph of a $C\sp 1$-function over $\Pi\cap B\sb (r\sb 0)$. The $C\sp 1$-norm depends on the dimension and $\sup\sb {B\sp +}|u|$. The result is extended to general subdomains of the unit ball with $C\sp 3$-boundary.