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Title of the thesis: Parabolic double-phase systems

Thesis defender: Lauri Särkiö
Opponent: Professor Ulisse Stefanelli, University of Vienna, Austria
Custos: Professor Juha Kinnunen, Aalto University School of Science

This dissertation studies nonlinear partial differential equations of parabolic double-phase type. Partial differential equations model the real world, allowing us to calculate the behavior of various phenomena up to some simplifying assumptions. For example, under the assumption of linear dependence between the difference in temperature and heat transfer, the diffusion of heat, or other corresponding quantity, is described by the well-known heat equation. The parabolic double-phase equation is a nonlinear extension of the heat equation describing diffusion in two phases depending on a weight function. Phenomena exhibiting this kind of behavior include electrorheological fluids whose viscosity profile changes radically in presence of an electric field. For example, such fluids have been applied in advanced clutches and braking systems of automobiles. 

Our results answer questions related to the regularity and existence of solutions to parabolic double-phase systems of equations. In addition, we focus on identifying the minimal assumptions on solutions that are required for establishing these regularity results. We consider the regularity of both solutions and the gradient of solutions. For bounded solutions we show local Hölder continuity and for the gradient of solutions we consider higher integrability. These results represent essentially optimal statements on regularity under certain assumptions. Although our research is theoretical in nature, the mathematical foundation developed stays relevant in more practical studies. Besides the main results, we advance general methods in fields of analysis including nonlinear partial differential equations, calculus of variations and harmonic analysis. Some key elements in the proofs are a parabolic phase analysis of the double-phase model, intrinsic geometries and the method of Lipschitz truncation.

Keywords: nonlinear partial differential equation, regularity theory, parabolic double-phase equation

Thesis available for public display 7 days prior to the defence at .