Geometric Theory of Sobolev Spaces

该计划的目标是以Sobolev空间理论为主要工具来研究几何分析中的各种问题。由于Sobolev空间的理论有着广泛的应用，因此本文的命题并不局限于一个狭窄的问题，而是从几何分析的更广泛的领域中提出了许多问题。其中包括：（1）Sobolev空间中极大函数的有界性;（2）Sobolev扩张域;（3）等周不等式、Sobolev不等式和与Korn不等式和Strauss的Sobolev不等式有关的向量情形下的截断方法之间的相互作用;（4）欧氏空间中域间Sobolev映射的几何性质（5）度量空间的双Lipschitz嵌入;（6）流形、多面体和度量空间之间Sobolev映射的Lipschitz逼近，并与空间的拓扑有关。(7)（8）流形间映射的Jacobian的哈代空间正则性及其在变分法正则性问题中的应用Sobolev空间理论是二十世纪数学中最伟大的发现之一。Sobolev空间理论是研究非线性偏微分方程的最重要的工具，无论是在理论方面还是在数值实现方面，尽管Sobolev空间理论是在30年代后期创建的，但近年来，该理论已经取得了重大突破，通过将应用扩展到纯几何学的新领域，如度量空间上的分析，几何群论或代数拓扑学以及应用数学领域，例如变分的非凸演算。该提案的目的是在各个方面继续这项调查。PI打算就与该提案密切相关的问题向博士生提供建议。

The goal of the proposal is to investigate variousproblems in Geometric Analysis with the theory of Sobolevspaces as a main tool. As the theory of Sobolev spaces hasan impressive range of applications, the proposaldoes not focus on a narrow problematics but rather, itaddresses a number of problems from a broader range of areas inGeometric Analysis. That includes study of:(1) boundedness of maximal functions in Sobolev spaces;(2) Sobolev extension domains;(3) interplay between isoperimetric inequality, Sobolev inequalityand the truncation method in the vectorial case related to Korn'sinequality and the Sobolev inequality of Strauss;(4) geometric properties of Sobolev mappings between domains in theEuclidean space (this research is related to problems in nonlinearelasticity);(5) bi-Lipschitz embeddings of metric spaces;(6) Lipschitz approximation of Sobolev mappings between manifolds,polyhedra and metric spaces with connections to the topology of spaces.(7) degree theory of mappings in Orlicz-Sobolev spaces in the casein which the target space is a rational homology sphere;(8) the Hardy space regularity of Jacobians of mappings between manifoldswith applications to the regularity questions in the calculus ofvariations.Theory of Sobolev spaces was one of the greatest discoveries inthe XXth century mathematics. This theory is the most important singletool in studying nonlinear partial differential equations, both in itstheoretical aspects and numerical implementation.Although the theory of Sobolev spaces has been created in the latethirties,in recent years, there have been major breakthroughs in the theory, byexpanding the applications to new areas of pure mathematicslike analysis on metric spaces, geometric group theory or algebraictopology as well as to areas in applied mathematics, like for example to non-convexcalculus of variations. The aim of the proposal is to continue thisinvestigation in its various aspects. The PI has intention to advise PhDstudents on problems closely related to the proposal.