Boundary regularity and compactness for overdetermined problems
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 4, p. 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: $\Delta u\left(x\right)={\chi }_{{}_{\Omega }}\left(x\right)f\left(x\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}D,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0\in \partial \Omega ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}u=|\nabla u|=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\Omega }^{c},$ 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

Classification:  35R35
@article{ASNSP_2003_5_2_4_787_0,
author = {Blank, Ivan and Shahgholian, Henrik},
title = {Boundary regularity and compactness for overdetermined problems},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 2},
number = {4},
year = {2003},
pages = {787-802},
zbl = {1170.35484},
mrnumber = {2040643},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2003_5_2_4_787_0}
}

Blank, Ivan; Shahgholian, Henrik. Boundary regularity and compactness for overdetermined problems. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 4, pp. 787-802. http://www.numdam.org/item/ASNSP_2003_5_2_4_787_0/

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