In this thesis we study the regularity of solutions for generalizations of the parabolic p-Laplace 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 Cauchy-Dirichlet problem. As a by-product 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 two-phase Stefan problem and show that there exists a solution to the Cauchy-Dirichlet 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 p-Laplacian.
The employed methods are mostly based on similar ideas to the ones typically used for the p-Laplace 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.
- , Supervisor
- Tuomo Kuusi, Advisor
|Publication status||Published - 2018|
|MoE publication type||G5 Doctoral dissertation (article)|
- partial differential equations, nonlinear analysis, regularity theory, parabolic equations, p-Laplace equation, Orlicz spaces, general growth conditions, regularity of solutions, obstacle problem, phase transtition, Stefan problem, boundary regularity, method of intrinsic scaling