Topics in the Theory of Elliptic Boundary Value Problems

这项数学研究项目集中在调和分析和偏微分方程问题上，重点研究二阶和高阶算子的椭圆型边值问题以及非光滑区域上系统的椭圆型边值问题。非光滑区域指的是具有孤立奇点或Lipschitz域的区域。虽然拉普拉斯方程的椭圆型边值问题已有二十多年的研究历史，但对于二阶椭圆组和高阶椭圆型方程的理论还不完善。另一个尚未完全理解的经典Dirichlet和Neumann边值问题的推广是混合边值问题。这个项目解决的公开问题分为以下几个研究主题：高阶椭圆算子的边值问题；Besov空间上的谱半径猜想；混合问题在Lipschitz区域上的适定性。这个数学研究项目的动机是数学物理和工程中自然产生的问题。在这一点上，提出问题的非光滑设置是基本的，因为大多数现实的物理模型涉及不规则的域。例如，高阶椭圆算子的边值问题在模拟扁壳、梁弯曲和固定板的背景下具有工程应用。混合边值问题模拟了几个物理量的行为，如冶金熔化过程中的温度、被加热物体冲压或冲压的弹性固体的热弹性势，或通过多孔材料的渗流。这项数学研究项目将对上述学科做出贡献，并产生新的数学技术。在追求上述研究方向的同时，OTT将发起旨在增加妇女和其他代表性不足群体对数学的参与的活动。这些活动将包括与当地初中和高中的外联活动，以及为本科生和研究生提供的研究和网络机会。

This mathematics research project focuses on problems in harmonic analysis and partial differential equations, with an emphasis on the study of elliptic boundary value problems for second and higher order operators and for systems in non-smooth domains. A non-smooth domain refers to a domain with either isolated singularities or a Lipschitz domain. While elliptic boundary value problems for the Laplacian have been well understood for over twenty years, the theory for second order elliptic systems and higher order elliptic equations is incomplete. Another generalization of the classical Dirichlet and Neumann boundary value problems that is not yet fully understood is the mixed boundary value problem. This project addresses open problems that are categorized by the following research themes: boundary value problems for higher order elliptic operators; the spectral radius conjecture on Besov spaces; well-posedness of the mixed problem in Lipschitz domains. This mathematics research project is motivated by problems that naturally arise in mathematical physics and engineering. In this regard, the non-smooth setting in which the problems are posed is fundamental since most realistic physical models involve irregular domains. For example, boundary value problems for higher order elliptic operators have applications to engineering in the context of modeling shallow shells, beam bending, and clamped plates. Mixed boundary value problems model the behavior of several physical quantities such as the temperature in a metallurgical melting process, the thermo-elastic potential of an elastic solid punched or stamped by a heated object, or the seepage through a porous material. This mathematics research project will contribute to the aforementioned disciplines as well as produce new mathematical techniques. While pursuing the research directions outlined above, Ott will initiate activities aimed at increasing the participation of women and other under-represented groups in mathematics. These activities will include outreach activities with local middle and high schools, and research and networking opportunities for undergraduate and graduate students.