Existence and regularity for parabolic quasi minimizers on metric measure spaces

本项目的目的是对一般度量测度空间上非线性抛物极小化问题解的存在性和正则性理论作出实质性的贡献。本文旨在建立一种系统的方法来推广A. Grigor'yan和L. Saloff-Coste关于黎曼流形上热方程解与Harnack估计有效性之间的关系。有趣的是将这个结果推广到两个方面：首先，我们打算考虑度量测度空间，支持测度的加倍性质和庞加莱不等式，而不是黎曼流形。其次，而不是（线性）热方程，非线性问题应调查。该项目的主要困难在于一方面在度量测度空间的非常一般的概念，另一方面在非线性的偏微分方程和积分泛函正在考虑。在一般的度量测度空间中，不可能谈论“方向”或“分部积分”，因此缺乏适当形式的导数。这使得有必要与所谓的“上梯度”，这是根据定义的特征的Sobolev函数在Euklidean空间的幂p可积向量场。这个概念不允许引入偏微分方程的合理定义，但它有助于将最小化问题推广到度量测度空间。非线性的问题造成了一些进一步严重的困难，这基本上是已知的非线性抛物型微分方程在n维Euklidean空间的理论。这些困难必须在纯最小化问题的水平上被克服-没有任何相关的偏微分方程。

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.