Higher regularity of free boundaries in obstacle problems

Autor/a

Kukuljan, Teo

Director/a

Ros, Xavier

Data de defensa

2022-10-07

Pàgines

209 p.



Departament/Institut

Universitat de Barcelona. Departament de Matemàtiques i Informàtica

Resum

In the thesis we consider higher regularity of the free boundaries in different variations of the obstacle problem, that is, when the Laplace operator b. is replaced with another elliptic or parabolic operator. In the fractional obstacle problem with drift (L = (-'6.)8 + b · v'), we prove that for constant b, and irrational s > ½ the free boundary is C00 near regular points as long as the obstacle is C00. To do so we establish higher order boundary Harnack inequalities for linear equations. This gives a bootstrap argument, as the normal of the free boundary can be expressed with quotients of derivatives of solution to the obstacle problem. Furthermore we establish the boundary Harnack estímate for linear parabolic operators (L = Ot - b.) in parabolic C1 and C1•°' domains and give a new proof of the higher order boundary Harnack estímate in ck,a domains. In the similar way as in the fractional obstacle problem with drift this implies that the free boundary in the parabolic obstacle problem is C00 near regular points. We also study the regularity of the free boundary in the parabolic fractional obstacle problem (L = Ot + (-b.)8) in the cases > ½- We are able to provea boundary Harnack estímate in C1•°' domains, which improves the regularity of the free boundary from C1•°' to C2•°'. Finally, we establish the full regularity theory for free boundaries in fully non-linear parabolic obstacle problem. Concretely we find the splitting of the free boundary into regular and singular points, we show that near regular points the free boundary is locally a graph of a C00 function, and that the singular points are '' rare" - they can be covered with a Lipschitz manifold of co-dimension 2, which is arbitrarily flat in space.

Paraules clau

Equacions diferencials parabòliques; Ecuaciones diferenciales parabólicas; Parabolic differential equations; Equacions diferencials el·líptiques; Ecuaciones diferenciales elípticas; Elliptic differential equations; Càlcul; Cálculo; Calculus

Matèries

51 - Matemàtiques

Àrea de coneixement

Ciències Experimentals i Matemàtiques

Nota

Programa de Doctorat en Matemàtica i Informàtica

Documents

TK_PhD-THESIS.pdf

1.639Mb

 

Drets

L'accés als continguts d'aquesta tesi queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: http://creativecommons.org/licenses/by-nc-nd/4.0/
L'accés als continguts d'aquesta tesi queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: http://creativecommons.org/licenses/by-nc-nd/4.0/

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