Mathematical Sciences: Variable Coefficient Fourier Analysis

Sogge DMS-9424418 索格将继续他的工作，在经典分析和偏微分方程的问题，涉及到波动方程。 他将试图证明尖锐的结果有关傅立叶分析领域将有应用问题的非线性波动方程在有界域。 他还将研究欧几里德空间中半线性波动方程的小数据存在定理。 这将需要新类型的混合范数估计的波动方程的解决方案，涉及非平凡的角范数。 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。

Sogge DMS-9424418 Sogge will continue his work on problems in classical analysis and pde's that relate to the wave equation. He will attempt to prove sharp results concerning Fourier analysis in domains which will have applications to questions about nonlinear wave equations in bounded domains. He will also investigate small-data existence theorems for semilinear wave equations in Euclidean space. This will require new types of mixed-norm estimates for solutions of the wave equation involving non-trivial angular norms. 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.