Mathematical Sciences: Nonlinear Partial Differential Equations and Their Applications to Evolving Surfaces, Phase Transitions and Stochastic Control

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9500940 Soner This is a proposal to National Science Foundation to work on the applications of nonlinear partial differential equations to phase transitions, evolving surfaces and stochastic control. The proposed studies in evolutionary phase transitions and evolving surfaces concern the development of the asymptotic analysis of systems of reaction-diffusion equations (including the phase-field, Cahn-Hilliard, and the vector valued Ginzburg-Landau equations), the further study of the interface propagation with three or more phases, the analysis of the dynamics of surfaces with it arbitrary codimension, and the development of a level set approach for these surfaces. The proposed studies in stochastic control concern the analysis of mathematical financial models and the approximation of complex, controlled systems by simpler and more tractable continuum models. The theory of viscosity solutions have been very successful in establishing the connection between various approximate and the original models. Proposed here is to continue this research when the continuum models are singularly controlled. In mathematical finance, partial differential equations have been an efficient tool not only in the analysis but also in numerical computations. Option pricing provides a good example of this. The principal investigator proposes to use the theory of viscosity solutions in the analysis of models with transaction costs. %%% Research on the phase transitions and evolving surfaces is related to several models in materials science modeling the dynamics of grain boundaries, interfaces between different phases and defects. Understanding the propagation of defects and interfaces is of fundamental importance not only for its intrinsic interest but also for its technological importance. Research on stochastic control is related to problems in manufacturing, communications and mathematical finance. In mathematical finance, Black-Scholes type option pricing problems will be investigated in the presence of transaction costs. ***