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On the two-phase membrane problem with coefficients below the Lipschitz threshold
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-4309-9242
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-1316-7913
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2009 (English)In: Annales de l'Institut Henri Poincare. Analyse non linéar, ISSN 0294-1449, E-ISSN 1873-1430, Vol. 26, no 6, 2359-2372 p.Article in journal (Refereed) Published
Abstract [en]

We study the regularity of the two-phase membrane problem, with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the C-1,C-1-regularity of the solution and that the free boundary is, near the so-called branching points, the union of two C-1-graphs. In our case, the same monotonicity formula does not apply in the same way. In the absence of a monotonicity formula, we use a specific scaling argument combined with the classification of certain global solutions to obtain C-1,C-1-estimates. Then we exploit some stability properties with respect to the coefficients to prove that the free boundary is the union of two Reifenberg vanishing sets near so-called branching points.

Place, publisher, year, edition, pages
2009. Vol. 26, no 6, 2359-2372 p.
Keyword [en]
FREE-BOUNDARY PROBLEMS; OBSTACLE-PROBLEM; DIFFERENTIAL EQUATIONS; 2 PHASES; REGULARITY
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-10334DOI: 10.1016/j.anihpc.2009.03.006ISI: 000272561600014Scopus ID: 2-s2.0-71849090341OAI: oai:DiVA.org:kth-10334DiVA: diva2:214783
Note
QC20100621Available from: 2009-05-06 Created: 2009-05-06 Last updated: 2017-12-13Bibliographically approved
In thesis
1. Regularity properties of two-phase free boundary problems
Open this publication in new window or tab >>Regularity properties of two-phase free boundary problems
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers which are all related to the regularity properties of free boundary problems. The problems considered have in common that they have some sort of two-phase behaviour.In papers I-III we study the interior regularity of different two-phase free boundary problems. Paper I is mainly concerned with the regularity properties of the free boundary, while in papers II and III we devote our study to the regularity of the function, but as a by-product we obtain some partial regularity of the free boundary.The problem considered in paper IV has a somewhat different nature. Here we are interested in certain approximations of the obstacle problem. Two major differences are that we study regularity properties close to the fixed boundary and that the problem converges to a one-phase free boundary problem.

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. viii, 36 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 09:07
Keyword
Mathematics
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-10336 (URN)978-91-7415-288-3 (ISBN)
Public defence
2009-06-05, F3, Lindstedtsvägen 26, KTH, 14:00 (English)
Opponent
Supervisors
Note
QC 20100728Available from: 2009-05-26 Created: 2009-05-06 Last updated: 2010-07-28Bibliographically approved
2. Monotonicity formulas and applications in free boundary problems
Open this publication in new window or tab >>Monotonicity formulas and applications in free boundary problems
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of three papers devoted to the study of monotonicity formulas and their applications in elliptic and parabolic free boundary problems. The first paper concerns an inhomogeneous parabolic problem. We obtain global and local almost monotonicity formulas and apply one of them to show a regularity result of a problem that arises in connection with continuation of heat potentials.In the second paper, we consider an elliptic two-phase problem with coefficients bellow the Lipschitz threshold. Optimal $C^{1,1}$ regularity of the solution and a regularity result of the free boundary are established.The third and last paper deals with a parabolic free boundary problem with Hölder continuous coefficients. Optimal $C^{1,1}\cap C^{0,1}$ regularity of the solution is proven.

Place, publisher, year, edition, pages
Stockholm: KTH, 2010. 37 p.
Keyword
Partial differential equations, PDE, Free boundary problems, Monotonicity formulas
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-12405 (URN)978-91-7415-595-2 (ISBN)
Public defence
2010-05-07, F3, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC20100621Available from: 2010-04-20 Created: 2010-04-16 Last updated: 2010-06-21Bibliographically approved

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Publisher's full textScopushttp://dx.doi.org/10.1016/j.anihpc.2009.03.006

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Lindgren, ErikShahgholian, Henrik

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