# Boundary regularity and compactness for overdetermined problems

• Volume: 2, Issue: 4, page 787-802
• ISSN: 0391-173X

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## Abstract

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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:$\Delta u\left(x\right)={\chi }_{{}_{\Omega }}\left(x\right)f\left(x\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}D,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0\in \partial \Omega ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}u=|\nabla u|=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\Omega }^{c},$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$u\left(x\right)\equiv 0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left\{\phantom{\rule{0.277778em}{0ex}}\left({x}^{\text{'}},\phantom{\rule{0.277778em}{0ex}}{x}_{n}\right)\phantom{\rule{0.277778em}{0ex}}|\phantom{\rule{0.277778em}{0ex}}{x}_{n}=0\phantom{\rule{0.277778em}{0ex}}\right\}\phantom{\rule{0.166667em}{0ex}}.$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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