Mathematical Sciences: Harmonic Analysis and Partial Differential Equations

PI：Kenig DMS-9500725 凯尼格将继续他的研究发展的各个方面的谐波分析和理论应用偏微分方程。 主要重点将是研究最小光滑条件下的边值问题，边界唯一连续性和控制理论的“大爆炸”性质，两相自由边界问题的正则性，以及波传播中产生的非线性方程和系统的适定性。 必要的谐波分析方法的研究，这些问题将同时进行调查。 偏微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。

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.