Regularity for parabolic quasiminimizers in metric measure spaces

Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi Author Mathias Masson Name of the doctoral dissertation Regularity for parabolic quasiminimizers in metric measure spaces Publisher School of Science Unit Department of Mathematics and Systems Analysis Series Aalto University publication series DOCTORAL DISSERTATIONS 89/2013 Field of research Mathematical analysis Manuscript submitted 28 February 2013 Date of the defence 30 May 2013 Permission to publish granted (date) 29 April 2013 Language English Monograph Article dissertation (summary + original articles) Abstract In this thesis we study in the context of metric measure spaces, some methods which in Euclidean spaces are closely related to questions concerning regularity of nonlinear parabolic partial differential equations of the evolution p-Laplacian type and of the doubly nonlinear type. To be more specific, we are interested in methods which are based only on energy type estimates.In this thesis we study in the context of metric measure spaces, some methods which in Euclidean spaces are closely related to questions concerning regularity of nonlinear parabolic partial differential equations of the evolution p-Laplacian type and of the doubly nonlinear type. To be more specific, we are interested in methods which are based only on energy type estimates. We take a purely variational approach to parabolic partial differential equations, and use the concept of parabolic quasiminimizers together with upper gradients and Newtonian spaces, to develop regularity theory for nonlinear parabolic partial differential equations in the context of general metric measure spaces. The underlying metric measure space is assumed to be equipped with a doubling measure and to support a weak Poincaré inequality. We define parabolic quasiminimizers in metric measure spaces and establish some preliminary results. Then we prove several regularity results for parabolic quasiminimizers in metric measure spaces, using energy estimates and the properties of the underlying metric measure space. The results we present are previously unpublished. We prove local Hölder continuity in metric measure spaces for locally bounded parabolic quasiminimizers related to degenerate evolution p-Laplacian equations. We prove a scale and location invariant weak Harnack estimate in metric measure spaces for parabolic minimizers related to the doubly nonlinear equation in the general case, where p is strictly between one and infinity. We prove higher integrability results in metric measure spaces, both in the local case and up to the boundary, for parabolic quasiminimizers related to the heat equation. Lastly, we prove a comparison principle in metric measure spaces for parabolic superand subminimizers, and a uniqueness result for minimizers related to the evolution p-Laplacian equation in the general case, where p is strictly between one and infinity. The results and the methods used in the proofs are discussed in detail, and some related open questions are presented.