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Non-transversal intersection of free and fixed boundary for fully nonlinear elliptic operators in two dimensions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). (Harmonic analysis and PDE)ORCID iD: 0000-0002-9608-3984
(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
URN: urn:nbn:se:kth:diva-177147OAI: diva2:871584

QS 201511

Available from: 2015-11-16 Created: 2015-11-16 Last updated: 2015-12-10Bibliographically 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.
TRITA-MAT-A, 2015:14
free boundary, elliptic, fully non-linear
National Category
Mathematical Analysis
Research subject
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)
Swedish Research Council

QC 20151210

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

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