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Monotonicity formulas and applications in free boundary problems
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). (Analysis)
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 [en]
Partial differential equations, PDE, Free boundary problems, Monotonicity formulas
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-12405ISBN: 978-91-7415-595-2 (print)OAI: oai:DiVA.org:kth-12405DiVA: diva2:310874
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
List of papers
1. A parabolic almost monotonicity formula
Open this publication in new window or tab >>A parabolic almost monotonicity formula
2008 (English)In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 341, no 2, 429-454 p.Article in journal (Refereed) Published
Abstract [en]

We prove the parabolic counterpart of the almost monotonicity formula of Caffarelli, Jerison and Kening for pairs of functions u +/-( x, s) in the strip S-1 = R-n x (-1, 0] satisfying u +/- >= 0, (Delta - partial derivative(S))u +/- >= -1, u(+) center dot u(-) = 0 in S-1. We also establish a localized version of the formula as well as prove one of its variants. At the end of the paper we give an application to a free boundary problem related to the caloric continuation of heat potentials.

Keyword
FREE-BOUNDARY, REGULARITY
Identifiers
urn:nbn:se:kth:diva-13501 (URN)10.1007/s00208-007-0195-y (DOI)000254203700007 ()2-s2.0-41249085294 (Scopus ID)
Note
QC20100621Available from: 2010-06-21 Created: 2010-06-21 Last updated: 2010-06-21Bibliographically approved
2. On the two-phase membrane problem with coefficients below the Lipschitz threshold
Open this publication in new window or tab >>On the two-phase membrane problem with coefficients below the Lipschitz threshold
2009 (English)In: Annales de l'Institut Henri Poincare. Analyse non linéar, ISSN 0294-1449, 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.

Keyword
FREE-BOUNDARY PROBLEMS; OBSTACLE-PROBLEM; DIFFERENTIAL EQUATIONS; 2 PHASES; REGULARITY
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-10334 (URN)10.1016/j.anihpc.2009.03.006 (DOI)000272561600014 ()2-s2.0-71849090341 (Scopus ID)
Note
QC20100621Available from: 2009-05-06 Created: 2009-05-06 Last updated: 2012-04-14Bibliographically approved
3. Regularity of a parabolic free boundary problem with Hölder continuous coefficients
Open this publication in new window or tab >>Regularity of a parabolic free boundary problem with Hölder continuous coefficients
(English)Manuscript (preprint) (Other academic)
Identifiers
urn:nbn:se:kth:diva-13506 (URN)
Note
QC20100621Available from: 2010-06-21 Created: 2010-06-21 Last updated: 2010-06-21Bibliographically approved

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  • apa
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