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Regularity properties of two-phase free boundary problems
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0003-4309-9242
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 [en]
Mathematics
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-10336ISBN: 978-91-7415-288-3 (print)OAI: oai:DiVA.org:kth-10336DiVA: diva2:214797
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
List of papers
1. Regularity of the free boundary in a two-phase semilinear problem in two dimensions
Open this publication in new window or tab >>Regularity of the free boundary in a two-phase semilinear problem in two dimensions
2008 (English)In: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 57, no 7, 3397-3418 p.Article in journal (Refereed) Published
Abstract [en]

We study minimizers of the energy functional ∫D (|∇u|2 + 2(λ+(u+)p + λ-(u-)p)) dx for p ∈ (0, 1) without any sign restriction on the function u. The main result states that in dimension two the free boundaries Γ+ = ∂{u > 0} ∩ D and Γ- = ∂{u < 0} n D are C1 regular.

Keyword
Alexandrov reflection-comparison; Monotonicity formula; Regularity of the free boundary; Semilinear equation
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-10332 (URN)10.1512/iumj.2008.57.3433 (DOI)000263576400014 ()2-s2.0-62649101823 (Scopus ID)
Note
QC 20100728Available from: 2009-05-06 Created: 2009-05-06 Last updated: 2010-12-06Bibliographically approved
2. A two-phase obstacle-type problem for the p-Laplacian
Open this publication in new window or tab >>A two-phase obstacle-type problem for the p-Laplacian
2009 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 35, no 4, 421-433 p.Article in journal (Refereed) Published
Abstract [en]

We study the so-called two-phase obstacle-type problem for the p-Laplacian when p is close to 2. We introduce a new method to obtain the optimal growth of the function from branch points, i.e. two-phase points in the free boundary where the gradient vanishes. As a by-product we can locally estimate the (n - 1)-Hausdorff-measure of the free boundary for the special case when p > 2.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-10333 (URN)10.1007/s00526-008-0212-3 (DOI)000265084800002 ()2-s2.0-64749108515 (Scopus ID)
Note
QC 20100728Available from: 2009-05-06 Created: 2009-05-06 Last updated: 2010-12-03Bibliographically approved
3. 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
4. On the penalized obstacle problem in the unit half ball
Open this publication in new window or tab >>On the penalized obstacle problem in the unit half ball
2010 (English)In: Electronic Journal of Differential Equations, ISSN 1072-6691, no 9, 1-12 p.Article in journal (Refereed) Published
Abstract [en]

We study the penalized obstacle problem in the unit half ball,i.e. an approximation of the obstacle problem in the unit half ball. The mainresult states that when the approximation parameter is small enough and whencertain level sets are sufficiently close to the hyperplane {x1 = 0}, then theselevel sets are uniformly C1 regular graphs. As a by-product, we also recoversome regularity of the free boundary for the limiting problem, i.e., for theobstacle problem.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-10335 (URN)
Note
QC 20100728Available from: 2009-05-06 Created: 2009-05-06 Last updated: 2010-07-28Bibliographically approved

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