Mathematical Sciences: Elliptic and Parbolic Partial Differential Equations

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This project concentrates on three areas of mathematical research in partial differential equations. The first concerns the general theory of degenerate elliptic Bellman equations with underlying quasilinear operators. These equations arise as the dynamic programming equations for controlled diffusions. In many important cases the equations become degenerate, i.e. the governing matrix in the equation becomes singular, and the wealth of theory on elliptic equations is no longer applicable. The goals of this work are to develop new techniques for proving a prior estimates on solutions, investigation of nonlinear equations related to the real and complex Monge-Ampere equations and to study equations of surfaces with prescribed curvature. A second area of investigation focuses on viscosity solutions of nonlinear elliptic equations. The use of this concept, which is similar to that of finding the value function in control theory, allows one to investigate solutions of nonlinear equations without assuming that they are differentiable. Once a solution is known to exist in the viscosity sense, an important problem of numerical or other approximation arises. Work will be done developing methods for estimating the speed of convergence of approximations to the true solution. To construct good approximations one must also make further investigations into the smoothness properties of viscosity solutions. The third element of this research concerns properties of solutions of linear parabolic equations with white noise forcing terms. These are equations in which randomness is a natural ingredient. They arise in filtering problems, genetics, quantum mechanics and magneto-dynamics. Very little work has been done in approximating solutions to such equations. This research will concentrate on those equations defined on bounded domains in which the boundary conditions introduce unusual problems not encountered in the deterministic equations 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.