Elliptic Boundary Value Problems, Harmonic Analysis and Spectral Theory

该项目是针对粗糙区域上的椭圆型偏微分方程组理论中的一系列问题。最终目标是了解域的几何形状、数据的性质、方程的结构和解的规律性之间的复杂关系。在其他问题中，该项目解决了任意区域上的高阶椭圆方程解的基本性质(如最大值原理和Wiener准则)，根据存在边界奇点的数据对Dirichlet和Neumann问题的解的精确估计，以及具有复有界可测系数的椭圆算子。该研究计划融合了现代分析的不同分支(调和分析、算子和谱理论、函数空间)的技术，并促进了新方法的发展，这些方法揭示了一些全新的现象。椭圆问题自然而然地出现在物理学的各个分支中，如静电学、热力学和弹性力学。然而，尽管椭圆型偏微分方程的理论有很长的历史，但它包含了许多悬而未决的问题。这项工作将有助于在上述科学和工程领域取得进一步进展，拟议的研究成果将通过出版物和在国家和国际专业会议上的陈述、与分析和其他领域的研究人员的交流、正式和非正式的教育和外联活动，在不同层面传播。

The proposed project is aimed at a wide range of problems in the theory of elliptic partial differential equations on rough domains. The ultimate goal is to understand the intricate relations between the geometry of the domain, the nature of the data, the structure of the equation, and the regularity of the solutions. Among other problems, the project addresses fundamental properties of solutions to the higher order elliptic equations in arbitrary domains (such as maximum principle and Wiener criterion), sharp estimates on the solutions of Dirichlet and Neumann problems in terms of the data in the presence of boundary singularities, as well as elliptic operators with complex bounded measurable coefficients. The research plan incorporates techniques originating from different branches of modern analysis (harmonic analysis, operator and spectral theory, function spaces) and promotes the development of new methods, which unravel some completely new phenomena. The elliptic problems naturally arise in various branches of physics, such as electrostatics, thermodynamics, and elasticity. However, despite its long history, the theory of elliptic partial differential equations contains many open questions. This work will contribute to further progress in the aforementioned areas of science and engineering, and the results of the proposed research will be disseminated at various levels: through publications and presentations in national and international professional meetings, communication with researchers in analysis and other fields, formal and informal educational and outreach activities.