Boundary regularity and compactness for overdetermined problems

Blank, Ivan; Shahgholian, Henrik

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 787-802.

Let $D$ be either the unit ball $B_1\left(0\right)$ or the half ball $B_1^{+}\left(0\right),$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem:

$$\Delta u\left(x\right)={\chi }_{{}_{\Omega }}\left(x\right)f\left(x\right)\text{in}D,0\in \partial \Omega ,u=\left|\nabla u\right|=0\text{in}{\Omega }^c,$$

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 $D=B_1^{+}\left(0\right),$ then we impose in addition that

$$u\left(x\right)\equiv 0\text{on}\left\{(x^{\text{'}},x_n)\right|x_n=0\}.$$

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 (2000) lead to regularity of $\partial \Omega$ under a weaker thickness assumption

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