Boundary regularity and compactness for overdetermined problems
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Ivan Blank [1] ; Henrik Shahgholian [2]
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[1] University of Louisville
University of Louisville
Estados Unidos
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[2] Royal Institute of Tchnology Stockholm, Sweden
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- Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 2, Nº 4, 2003, págs. 787-802
- Idioma: inglés
- Enlaces
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Resumen
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
