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Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Carnegie Mellon University, USA.ORCID iD: 0000-0002-9608-3984
2016 (English)In: Annales de l'Institut Henri Poincare. Analyse non linéar, ISSN 0294-1449, E-ISSN 1873-1430, Vol. 33, no 5, 1259-1277 p.Article in journal (Refereed) Published
Abstract [en]

We consider fully nonlinear obstacle-type problems of the form. F(D2u,x)=f(x)a.e. in B1∩Ω,|D2u|≤Ka.e. in B1\Ω, where Ω is an open set and K>0. In particular, structural conditions on F are presented which ensure that W2,n(B1) solutions achieve the optimal C1,1(B1/2) regularity when f is Hölder continuous. Moreover, if f is positive on B-1, Lipschitz continuous, and u≠0⊂Ω, we obtain interior C1 regularity of the free boundary under a uniform thickness assumption on u=0. Lastly, we extend these results to the parabolic setting.

Place, publisher, year, edition, pages
Elsevier, 2016. Vol. 33, no 5, 1259-1277 p.
Keyword [en]
Nonlinear elliptic equations, Nonlinear parabolic equations, Free boundaries, Regularity theory, Obstacle problems
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-174721DOI: 10.1016/j.anihpc.2015.03.009ISI: 000383822400004Scopus ID: 2-s2.0-84929250256OAI: oai:DiVA.org:kth-174721DiVA: diva2:867436
Note

QC 20161017

Available from: 2015-11-05 Created: 2015-10-07 Last updated: 2017-12-01Bibliographically approved
In thesis
1. Non-linear Free Boundary Problems
Open this publication in new window or tab >>Non-linear Free Boundary Problems
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of an introduction and four research papers related to free boundary problems and systems of fully non-linear elliptic equations.

Paper A and Paper B prove optimal regularity of solutions to general elliptic and parabolic free boundary problems, where the operators are fully non-linear and convex. Furthermore, it is proven that the free boundary is continuously differentiable around so called "thick" points, and that the free boundary touches the fixed boundary tangentially in two dimensions.

Paper C analyzes singular points of solutions to perturbations of the unstable obstacle problem, in three dimensions. Blow-up limits are characterized and shown to be unique. The free boundary is proven to lie close to the zero-level set of the corresponding blow-up limit. Finally, the structure of the singular set is analyzed.

Paper D discusses an idea on how existence and uniqueness theorems concerning quasi-monotone fully non-linear elliptic systems can be extended to systems that are not quasi-monotone.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2015. x, 21 p.
Series
TRITA-MAT-A, 2015:14
Keyword
free boundary, elliptic, fully non-linear
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-178110 (URN)978-91-7595-795-1 (ISBN)
Public defence
2016-01-22, Kollegiesalen, Brinellvägen 8, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council
Note

QC 20151210

Available from: 2015-12-10 Created: 2015-12-07 Last updated: 2015-12-15Bibliographically approved

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Minne, Andreas

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