Public defence in Mathematics, M.Sc. (Tech) Kristian Moring

This thesis studies two different prototypes of nonlinear partial differential equations with porous medium type and p-growth structure. These equations can be interpreted as nonlinear generalizations of the heat equation modeling various physical phenomena such as gas flow in porous medium, heat conduction or water movement in soil. The thesis focuses on regularity properties of solutions as well as their gradients to these equations. Boundary regularity for the gradient is shown in terms of higher integrability for porous medium type equations. Moreover, we demonstrate that both solutions and their gradients are stable with respect to the parameter characterizing the equation. If there is an obstacle restricting the behavior of the solution, we show that the solution is continuous provided that the obstacle is sufficiently regular. The obstacle problem is closely connected to the concept of supersolutions, which we define in the thesis as functions obeying a comparison principle. We show that supersolutions according to this definition are divided into two mutually exclusive classes for which we give several characterizations. The results in the thesis show that both solutions and their gradients possess properties that are mathematically relevant. The proofs of our results require new mathematical techniques and their implementations, which could be expected to be useful in other contexts as well.

Opponent is Professor José Miguel Urbano, University of Coimbra, Portugal

Custos is Professor Juha Kinnunen, Aalto University School of Science, Department of Mathematics and Systems Analysis

Contact details of the doctoral candidate: [email protected]

The public defence will be organised on campus (Otakaari 4, lecture hall 216).

The doctoral thesis is publicly displayed 10 days before the defence in the publication archive Aaltodoc of Aalto University.

Electronic thesis