Geometric Theory of Sobolev Spaces

本提案的目的是研究几何分析中的各种问题，以sobolevspace理论为主要工具。由于Sobolev空间理论具有令人印象深刻的应用范围，该提案并不关注狭窄的问题，而是解决了几何分析中更广泛领域的许多问题。这包括研究：(1)Sobolev空间中极大函数的有界性；(2) Sobolev可拓域；(3)在Korn不等式和Strauss的Sobolev不等式相关的向量情况下，等周不等式、Sobolev不等式与截断法之间的相互作用；(4)欧几里得空间域间Sobolev映射的几何性质（本研究涉及非线性非弹性问题）；(5)度量空间的bi-Lipschitz嵌入；(6)流形、多面体和度量空间之间Sobolev映射的Lipschitz近似。(7)以有理同调球为目标空间时Orlicz-Sobolev空间中映射的度理论；(8)流形间映射的雅可比矩阵的Hardy空间正则性及其在变分学正则性问题中的应用。索博列夫空间理论是20世纪数学界最伟大的发现之一。该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20世纪30年代后期创建的，但近年来，该理论取得了重大突破，将其应用扩展到纯数学的新领域，如度量空间的分析、几何群论或代数拓扑，以及应用数学的领域，如非凸变分。该提案的目的是在各个方面继续进行这项调查。PI打算就与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.