Geometric Theory of Sobolev Spaces

该方案的目的是以Sobolve空间理论为主要工具，研究几何分析中的各种问题。由于Sobolev空间的理论有着令人印象深刻的广泛应用，该提议不是集中在狭隘的问题学上，而是在几何分析中解决了更广泛领域的一些问题。这包括：(1)极大函数在Sobolev空间中的有界性；(2)Sobolev延伸域；(3)等周不等式、Sobolev不等式与截断方法之间的相互作用；(4)欧氏空间中区域间Sobolev映射的几何性质(本研究涉及到非线性相对论问题)；(5)度量空间的双Lipschitz嵌入；(6)与空间拓扑有关的流形、多面体和度量空间之间的Sobolev映射的Lipschitz逼近；(7)目标空间为有理同调球面的Orlicz-Sobolv空间中映射的度理论；(8)流形之间映射的雅可比的Hardy空间正则性及其在变分中的正则性问题上的应用。这一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。虽然Sobolev空间理论是在三十年代末才创立的，但近年来，该理论在理论上取得了重大突破，将其应用于诸如度量空间的纯数学分析、几何群论或代数拓扑学等新领域，以及应用数学的领域，如非凸变分。该提案的目的是继续对其各个方面进行调查。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.