The aim of this project is to make a substantial contrubution to existence and regularity theory for solutions of nonlinear parabolic minimization problems on general metric measure spaces. It is intended to establish a systematic approach to the generalization of a result by A. Grigor'yan and L. Saloff-Coste on the relation between solutions of the heat equation on Riemannian manifolds and the validity of Harnack estimates. The interest is to generalize this result in two ways: Firstly, one intends to consider metric measure spaces, supporting a doubling property of the measure and a Poincaré inequality, instead of a Riemannian manifold. Secondly, instead of the (linear) heat equation, nonlinear problems should be investigated.Main difficulties of the project consist on one hand in the very general concept of metric measure spaces, on the other hand in the nonlinearity of the partial differential equations and integral functionals under consideration. On general metric measure spaces it is not possible to speak of "direction" or "integration by parts" and consequently there's a lack of a suitable form of derivative. This makes it necessary to work with so-called "upper gradients" which are defined according to a characterization of Sobolev functions in the Euklidean space by to the power p integrable vector fields. This concept does not allow to introduce a reasonable definition of a partial differential equation, however it helps to generalize minimization problems to the context of metric measure spaces. The nonlinearity of the problem causes a number of further severe difficulties, which are basically already known from the theory of nonlinear parabolic differential equations in the n dimensional Euklidean space. These difficulties have to be overcome on the level of pure minimization problems -- without any associated partial differential equation behind.
