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Boundary Regularity and Compactness for Overdetermined Problems
KTH, Superseded Departments, Mathematics.ORCID iD: 0000-0002-1316-7913
2003 (English)In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 2, no 4, 787-802 p.Article in journal (Refereed) Published
Abstract [en]

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 of D solve the following overdetermined problem: Delta u (x) = chi(Omega) (x) f (x) in D, 0 is an element of partial derivative Omega, u = vertical bar del u vertical bar = 0 in Omega(c), where chi(Omega) denotes the characteristic function of Omega, Omega(c) denotes the set D\Omega, and the equation is satisfied in the sense of distributions. When D = B-1(+)(0), then we impose in addition that u(x) 0 {(x', x(n))vertical bar x(n) = 0} We show that a fairly mild thickness assumption on Omega(c) will ensure enough compactness on it to give us "blow-up" limits, and we show how this compactness leads to regularity of partial derivative Omega. In the case where f is positive and Lipschitz, the methods developed in Caffarelli, Karp, and Shahgholian (2000) lead to regularity of partial derivative Omega under a weaker thickness assumption.

Place, publisher, year, edition, pages
2003. Vol. 2, no 4, 787-802 p.
Identifiers
URN: urn:nbn:se:kth:diva-23237ISI: 000207016700006OAI: oai:DiVA.org:kth-23237DiVA: diva2:341935
Note
QC 20100525Available from: 2010-08-10 Created: 2010-08-10 Last updated: 2017-12-12Bibliographically approved

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Shahgholian, Henrik

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