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Non-linear Free Boundary Problems
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). (Harmonic analysis and PDE)ORCID iD: 0000-0002-9608-3984
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 [en]
free boundary, elliptic, fully non-linear
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-178110ISBN: 978-91-7595-795-1 (print)OAI: oai:DiVA.org:kth-178110DiVA: diva2:877717
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
List of papers
1. Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems
Open this publication in new window or tab >>Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems
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
Keyword
Nonlinear elliptic equations, Nonlinear parabolic equations, Free boundaries, Regularity theory, Obstacle problems
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-174721 (URN)10.1016/j.anihpc.2015.03.009 (DOI)000383822400004 ()2-s2.0-84929250256 (Scopus ID)
Note

QC 20161017

Available from: 2015-11-05 Created: 2015-10-07 Last updated: 2017-12-01Bibliographically approved
2. Non-transversal intersection of free and fixed boundary for fully nonlinear elliptic operators in two dimensions
Open this publication in new window or tab >>Non-transversal intersection of free and fixed boundary for fully nonlinear elliptic operators in two dimensions
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In the study of classical obstacle problems, it is well known that in many configurations the free boundary intersects the fixed boundary tangentially. The arguments involved in producing results of this type rely on the linear structure of the operator. In this paper we employ a different approach and prove tangential touch of free and fixed boundary in two dimensions for fully nonlinear elliptic operators. Along the way, several n-dimensional results of independent interest are obtained such as BMO-estimates, C1,1 regularity up to the fixed boundary, and a description of the behavior of blow-up solutions.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-177147 (URN)
Note

QS 201511

Available from: 2015-11-16 Created: 2015-11-16 Last updated: 2015-12-10Bibliographically approved
3. Classification of Singularities in Unstable Free Boundary Problems
Open this publication in new window or tab >>Classification of Singularities in Unstable Free Boundary Problems
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-177150 (URN)
Note

QS 201511

Available from: 2015-11-16 Created: 2015-11-16 Last updated: 2016-02-11Bibliographically approved
4. Viscosity Solutions of Balanced Quasi-Monotone Fully Nonlinear Weakly Coupled Systems
Open this publication in new window or tab >>Viscosity Solutions of Balanced Quasi-Monotone Fully Nonlinear Weakly Coupled Systems
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-177149 (URN)
Note

QP 201511

Available from: 2015-11-16 Created: 2015-11-16 Last updated: 2015-12-10Bibliographically approved

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