Mathematical Sciences: Linear and Nonlinear Partial Differential Equations

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9400978 Tataru This award supports mathematical research on problems arising in the theory of partial differential equations. Several projects are to be considered. The first is a continuation of work on viscosity solutions for fully nonlinear first and second order Hamilton-Jacobi equations with unbounded Hamiltonian in infinite dimensions. The goal is to build upon the earlier work to consider issues such as the existence of discontinuous solutions, equations arising in dynamic programming corresponding to optimal control or differential games and second order problems in infinite dimensions. Other work will focus on the study of unique continuation problems for solutions to linear partial differential equations and the problem of determining the minimal regularity of the coefficients for which such results can be obtained. A final goal is that of obtaining a complete characterization of the essential spectrum for semigroups generated by hyperbolic boundary value problems with applications in stabilization. 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. ***