Nonlinear variational problems on metric measure spaces

This dissertation studies existence and regularity properties of functions related to the calculus of variations on metric measure spaces that support a weak Poincaré inequality and doubling measure. The work consists of four articles in which we study the total variation flow and quasiminimizers of a (p,q)-Dirichlet integral. More specifically, we define variational solutions to the total variation flow in metric measure spaces. We establish existence and, using energy estimates and the properties of the underlying metric, we give necessary and sufficient conditions for a variational solution to be continuous ata given point. We then take a purely variational approach to a (p,q)-Dirichlet integral, define its quasiminimizers, and using the concept of upper gradients together with Newtonian spaces, we develop interior regularity, as well as regularity up to the boundary, in the context of general metric measure spaces.For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Hölder continuous and they satisfy Harnack inequality. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity and aWiener type regularity condition for continuity up to the boundary. Lastly, we prove higher integrability and stability results in metric measure spaces, for quasiminimizers related to the (p,q)-Dirichlet integral. The results and the methods used in the proofs are discussed in detail, and some related open questions are presented