Regularity for nonlinear parabolic partial differential equations
Research output: Thesis › Doctoral Thesis › Collection of Articles
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Abstract
In this thesis we study the regularity of solutions for generalizations of the parabolic pLaplace equation. The main focus is on equations with Orlicz type growth conditions for which we prove various regularity results, such as local boundedness of both weak solutions and their gradients. Moreover, we show the existence of a continuous solution up to the boundary for the CauchyDirichlet problem. As a byproduct we develop new approaches and techniques to handle the difficult nature of the equation that can be both degenerate and singular simultaneously. We also apply the obtained results to prove the existence of a unique solution to the related obstacle problem, and moreover, we show that in case the obstacle is continuous, the solution is as well.
The thesis also contains a section on phase transition problems. More precisely, we study the degenerate twophase Stefan problem and show that there exists a solution to the CauchyDirichlet problem that is continuous up to the boundary. Moreover, we derive an explicit modulus of continuity at the boundary. The main difficulty stems from the additional degeneracy caused by the jump at the transition point. This is overcome by considering the equation in three different intrinsic geometries instead of the usual one for the pLaplacian.
The employed methods are mostly based on similar ideas to the ones typically used for the pLaplace equation, for example De Giorgi's method is applied in many of the proofs. However, due to the generality of the equations, it has been necessary to also find some new tools and ideas.
Details
Translated title of the contribution  Epälineaaristen parabolisten osittaisdifferentiaaliyhtälöiden säännöllisyys 

Original language  English 
Qualification  Doctor's degree 
Awarding Institution  
Supervisors/Advisors 

Publisher 

Print ISBNs  9789526079271 
Electronic ISBNs  9789526079288 
Publication status  Published  2018 
MoE publication type  G5 Doctoral dissertation (article) 
 partial differential equations, nonlinear analysis, regularity theory, parabolic equations, pLaplace equation, Orlicz spaces, general growth conditions, regularity of solutions, obstacle problem, phase transtition, Stefan problem, boundary regularity, method of intrinsic scaling
Research areas
ID: 30202733