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The singular set for the composite membrane problem
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-1316-7913
2007 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 271, no 1, 93-101 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we study the behavior of the singular set {u = vertical bar del u vertical bar = 0}, for solutions u to the free boundary problem Delta u = f chi{u >= 0} - g chi{u < 0}, with f > 0, f (x) + g(x) < 0, and f, g is an element of C-alpha. Such problems arise in an eigenvalue optimization for composite membranes. Here we show that if for a singular point z is an element of {u =del u = 0}, there are r(0) > 0, and c(0) > 0 such that the density assumption vertical bar{u < 0} boolean AND B-r (z)vertical bar >= c(0)r(2), for all r < r(0), holds, then z is isolated. The density assumption can be motivated by the following example: u = x(1)(2), f = 2, g < -2, and {u < 0} = theta.

Place, publisher, year, edition, pages
2007. Vol. 271, no 1, 93-101 p.
Keyword [en]
free-boundary problems, 2 phases
URN: urn:nbn:se:kth:diva-16396DOI: 10.1007/s00220-006-0160-8ISI: 000244303700003ScopusID: 2-s2.0-33847222168OAI: diva2:334438
QC 20100525Available from: 2010-08-05 Created: 2010-08-05Bibliographically approved

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Shahgholian, Henrik
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