Mathematical Sciences: Variable Coefficient Fourier Analysis

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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.