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Analysis of a free boundary at contact points with Lipschitz data
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0002-1316-7913
2015 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 367, no 7, 5141-5175 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we consider a minimization problem for the functional J(u) = ∫ B+<inf>1</inf> |∇u|2 + λ2 <inf>+</inf>χ<inf>{u>0}</inf> + λ2 <inf>−</inf>χ<inf>{u≤0}</inf> in the upper half ball B+ <inf>1</inf> ⊂ ℝn, n ≥ 2, subject to a Lipschitz continuous Dirichlet data on ∂B+ <inf>1</inf>. More precisely we assume that 0 ∈ ∂{u > 0} and the derivative of the boundary data has a jump discontinuity. If 0 ∈ ∂({u > 0} ∩ B+ <inf>1</inf>), then (for n = 2 or n ≥ 3 and the one-phase case) we prove, among other things, that the free boundary ∂{u > 0} approaches the origin along one of the two possible planes given by γx<inf>1</inf> = ±x<inf>2</inf>, where γ is an explicit constant given by the boundary data and λ± the constants seen in the definition of J(u). Moreover the speed of the approach to γx<inf>1</inf> = x<inf>2</inf> is uniform.

Place, publisher, year, edition, pages
2015. Vol. 367, no 7, 5141-5175 p.
Keyword [en]
Contact points, Free boundary problem, Regularity
National Category
URN: urn:nbn:se:kth:diva-167776DOI: 10.1090/S0002-9947-2015-06187-XISI: 000357045700022ScopusID: 2-s2.0-84927613989OAI: diva2:814274

QC 20150526

Available from: 2015-05-26 Created: 2015-05-22 Last updated: 2015-11-27Bibliographically approved

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Shahgholian, Henrik
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