# Boundary regularity and compactness for overdetermined problems

Ivan Blank; Henrik Shahgholian

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

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

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topBlank, Ivan, and Shahgholian, Henrik. "Boundary regularity and compactness for overdetermined problems." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.4 (2003): 787-802. <http://eudml.org/doc/84519>.

@article{Blank2003,

abstract = {Let $D$ be either the unit ball $B_1(0)$ or the half ball $B_1^+(0),$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem:\[ \Delta u(x) = \chi \_\{\_\Omega \}(x) f(x) \ \ \text\{in\} \ \ D, \ \ \ \ 0 \in \partial \Omega , \ \ \ \ u = |\nabla u| = 0 \ \ \text\{in\} \ \ \Omega ^c, \]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 $D = B_1^+(0),$ then we impose in addition that\[ u(x) \equiv 0 \ \ \text\{on\} \ \ \lbrace \; (x^\{\prime \}, \; x\_n) \; | \; x\_n = 0 \; \rbrace \,. \]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 (2000) lead to regularity of $\partial \Omega $ under a weaker thickness assumption},

author = {Blank, Ivan, Shahgholian, Henrik},

journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},

language = {eng},

number = {4},

pages = {787-802},

publisher = {Scuola normale superiore},

title = {Boundary regularity and compactness for overdetermined problems},

url = {http://eudml.org/doc/84519},

volume = {2},

year = {2003},

}

TY - JOUR

AU - Blank, Ivan

AU - Shahgholian, Henrik

TI - Boundary regularity and compactness for overdetermined problems

JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

PY - 2003

PB - Scuola normale superiore

VL - 2

IS - 4

SP - 787

EP - 802

AB - Let $D$ be either the unit ball $B_1(0)$ or the half ball $B_1^+(0),$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem:\[ \Delta u(x) = \chi _{_\Omega }(x) f(x) \ \ \text{in} \ \ D, \ \ \ \ 0 \in \partial \Omega , \ \ \ \ u = |\nabla u| = 0 \ \ \text{in} \ \ \Omega ^c, \]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 $D = B_1^+(0),$ then we impose in addition that\[ u(x) \equiv 0 \ \ \text{on} \ \ \lbrace \; (x^{\prime }, \; x_n) \; | \; x_n = 0 \; \rbrace \,. \]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 (2000) lead to regularity of $\partial \Omega $ under a weaker thickness assumption

LA - eng

UR - http://eudml.org/doc/84519

ER -

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