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Regularity results in free boundary problems
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
2016 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
##### Abstract [en]

This thesis consists of three scientific papers, devoted to the regu-larity theory of free boundary problems. We use iteration arguments to derive the optimal regularity in the optimal switching problem, and to analyse the regularity of the free boundary in the biharmonic obstacle problem and in the double obstacle problem.In Paper A, we study the interior regularity of the solution to the optimal switching problem. We derive the optimal C1,1-regularity of the minimal solution under the assumption that the zero loop set is the closure of its interior.In Paper B, assuming that the solution to the biharmonic obstacle problem with a zero obstacle is suÿciently close-to the one-dimensional solution (xn)3+, we derive the C1,-regularity of the free boundary, under an additional assumption that the noncoincidence set is an NTA-domain.In Paper C we study the two-dimensional double obstacle problem with polynomial obstacles p1 p2, and observe that there is a new type of blow-ups that we call double-cone solutions. We investigate the existence of double-cone solutions depending on the coeÿcients of p1, p2, and show that if the solution to the double obstacle problem with obstacles p1 = −|x|2 and p2 = |x|2 is close to a double-cone solution, then the free boundary is a union of four C1,-graphs, pairwise crossing at the origin.

##### sted, utgiver, år, opplag, sider
KTH: KTH Royal Institute of Technology, 2016. , s. 122
##### Serie
TRITA-MAT-A ; 2016-10
Matematik
##### Identifikatorer
OAI: oai:DiVA.org:kth-195178DiVA, id: diva2:1044289
##### Disputas
2016-12-02, D3, Kungl Tekniska Högskolan, Lindstedtsvägen 5, Stockholm, 13:00 (engelsk)
##### Veileder

QC 20161103

Tilgjengelig fra: 2016-11-03 Laget: 2016-11-02 Sist oppdatert: 2016-11-16bibliografisk kontrollert
##### Delarbeid
1. Optimal regularity in the optimal switching problem
Åpne denne publikasjonen i ny fane eller vindu >>Optimal regularity in the optimal switching problem
2016 (engelsk)Inngår i: Annales de l'Institut Henri Poincare. Analyse non linéar, ISSN 0294-1449, E-ISSN 1873-1430Artikkel i tidsskrift (Fagfellevurdert) In press
##### Abstract [en]

In this article we study the optimal regularity for solutions to the following weakly coupled system with interconnected obstacles{min⁡(−Δu1+f1,u1−u2+ψ1)=0min⁡(−Δu2+f2,u2−u1+ψ2)=0 arising in the optimal switching problem with two modes. We derive the optimal C1,1-regularity for the minimal solution under the assumption that the zero loop set L:={ψ1+ψ2=0} is the closure of its interior. This result is optimal and we provide a counterexample showing that the C1,1-regularity does not hold without the assumption L=L0‾.

Elsevier, 2016
##### Identifikatorer
urn:nbn:se:kth:diva-195175 (URN)10.1016/j.anihpc.2015.06.001 (DOI)000389108400003 ()2-s2.0-84939510853 (Scopus ID)

QC 20161116

Tilgjengelig fra: 2016-11-02 Laget: 2016-11-02 Sist oppdatert: 2019-10-02bibliografisk kontrollert
2. Regularity of the free boundary in the biharmonic obstacle problem
Åpne denne publikasjonen i ny fane eller vindu >>Regularity of the free boundary in the biharmonic obstacle problem
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
##### Identifikatorer
urn:nbn:se:kth:diva-195176 (URN)

QC 20161116

Tilgjengelig fra: 2016-11-02 Laget: 2016-11-02 Sist oppdatert: 2016-11-16bibliografisk kontrollert
3. Analysis of blow-ups for the double obstacle problem in dimension two
Åpne denne publikasjonen i ny fane eller vindu >>Analysis of blow-ups for the double obstacle problem in dimension two
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
##### Identifikatorer
urn:nbn:se:kth:diva-195177 (URN)

QC 20161116

Tilgjengelig fra: 2016-11-02 Laget: 2016-11-02 Sist oppdatert: 2016-11-16bibliografisk kontrollert

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