Existence and regularity for parabolic quasi minimizers on metric measure spaces

本课题的目的是对一般度量空间上的非线性抛物型极小化问题解的存在性和正则性理论作实质性的研究。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估计的有效性之间的关系的一个结果。我们的兴趣是通过两种方式来推广这一结果：第一，打算考虑度量空间，它支持度量的加倍性质和Poincaré不等式，而不是黎曼流形。其次，应该研究非线性问题，而不是(线性)热方程。该项目的主要困难一方面在于度量空间的非常一般的概念，另一方面在于所考虑的偏微分方程组和积分泛函的非线性。在一般度量空间上，不可能谈到“方向”或“部分积分”，因此缺乏适当形式的导数。这使得有必要处理所谓的“上梯度”，它是根据欧氏空间中的Soblev函数的特征通过p次方可积向量场来定义的。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度量空间的背景下。问题的非线性导致了一些更严重的困难，这些困难基本上已经从n维欧氏空间中的非线性抛物型微分方程理论中知道了。这些困难必须在纯极小化问题的水平上克服--后面没有任何相关的偏微分方程式。

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.