Abstract

Let $D$ be either the unit ball $B_1$ or the half ball $B_1^+,$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem: $$ \Delta u = \chi _{_\Omega } f \ \ \text{in} \ \ D, \ \ \ \ 0 \in \partial \Omega, \ \ \ \ u = |\nabla u| = 0 \ \ \text{in} \ \ \Omega ^c, $$ where $\chi _{_\Omega }$ denotes the characteristic function of $\Omega,$ $\Omega ^c$ denotes the set $D \setminus \Omega,$ and the equation is satisfied in the sense of distributions. When $D = B_1^+,$ then we impose in addition that $$ u \equiv 0 \ \ \text{on} \ \ \lbrace \; \; | \; x_n = 0 \; \rbrace \,. $$ We show that a fairly mild thickness assumption on $\Omega ^c$ will ensure enough compactness on $u$ to give us “blow-up” limits, and we show how this compactness leads to regularity of $\partial \Omega.$ In the case where $f$ is positive and Lipschitz, the methods developed in Caffarelli, Karp, and Shahgholian lead to regularity of $\partial \Omega $ under a weaker thickness assumption