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Joint Applied Math and Probability Seminar
Building Solutions to Nonlinear Elliptic and Parabolic Partial
Differential Equations
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Nonlinear elliptic and parabolic partial differential equations (PDEs) appear in problems from science, engineering, atmospheric/ocean studies, image processesing, and mathematical finance. The theory of viscosity solutions has been enormously successful in addressing the problems of existence, uniqueness, and stability for a wide class of such equations. A problem which has not been addressed with as much success is the construction of solutions. In some cases, exact solutions formulas exist, but for the most part, solutions must be found numerically. In the sprit of the classical 1928 paper of Courant, Freidrichs, and Lewy which used the finite difference method to construct solutions of linear PDEs, we construct solutions to nonlinear degenerate elliptic and parabolic PDEs. We show that provided a simple structure condition holds, the nonlinear finite difference schemes have unique solutions which satisfy a discrete comparison principle. We will present example schemes and computational results, for: valuation in math finance, motion by mean curvature, (which leads to a problem of minimization of the l1 norm) and the infinity Laplacian (which leads to a problem of minimization of the l infinity norm). |