Mathematical Sciences: Harmonic Analysis and Partial Differential Equations

凯尼格DMS-9500725凯尼格将继续研究调和分析的各个方面的发展和偏微分方程式的理论应用。重点研究极小光滑性条件下的边值问题，边界唯一延拓和控制理论的“大爆炸”性质，两相自由边界问题的正则性，以及波传播中所产生的非线性方程和系统的适定性。同时还将研究研究这些问题所需的调和分析方法。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。

PI: Kenig DMS-9500725 Kenig will continue his research on the development of various aspects of harmonic analysis and theory application to partial differential equations. The main emphasis will be on the study of boundary value problems under minimal smoothness conditions, boundary unique continuation and the "big-bang" property of control theory, regularity of two-phase free boundary problems, and the well-posedness of nonlinear equations and systems arising in wave propagation. The necessary harmonic analysis methods for the study of these problems will be investigated concurrently. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.