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A parabolic almost monotonicity formula
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Purdue University.
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.

Place, publisher, year, edition, pages
2008. Vol. 341, no 2, 429-454 p.
Keyword [en]
FREE-BOUNDARY, REGULARITY
Identifiers
URN: urn:nbn:se:kth:diva-13501DOI: 10.1007/s00208-007-0195-yISI: 000254203700007Scopus ID: 2-s2.0-41249085294OAI: oai:DiVA.org:kth-13501DiVA: diva2:325886
Note
QC20100621Available from: 2010-06-21 Created: 2010-06-21 Last updated: 2017-12-12Bibliographically approved
In thesis
1. 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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