Viscosity solution methods in partial differential equations and applications

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The proposal concentrates on the analysis of certain classes of nonlinear partial differential equations and their applications. Linking them together is the notion of viscosity solution. Part of the proposal focuses on equations of Hamilton-Jacobi-Bellman (HJB) type that are related to optimal control of stochastic partial differential equations. The HJB equations associated with their control are equations in infinite dimensional spaces. The theory of such equations is not well developed. The principal investigator (PI) studies them in the project paying special attention to several equations related to problems of particular interest. One of such problems is optimal control of fluid flow that can be reformulated as optimal control of deterministic or stochastic Navier-Stokes equations. Another problem comes from mathematical finance and is related to option pricing. It includes analysis of infinite dimensional equivalent of ``Black-Scholes" equation and its nonlinear version, so called ``Black-Scholes-Barenblatt" equation. HJB equations in Hilbert (or Banach) spaces are the key to the dynamic programming analysis of optimal control problems of systems driven by partial differential equations.These HJB equations must be investigated from the point of view of generalized solutions. Viscosity solutions should provide the right approach to such equations and the proposed research should be an important ingredient in setting the stage for optimal control of infinite dimensional stochastic systems. The PI also proposes to investigate a class of fully nonlinear non-divergence form uniformly elliptic equations that includes generalizations of quasilinear equations and certain equations of geometric type, an important class in the elliptic theory. Such equations have not been studied systematically, especially when they are discontinuous in the spatial variable. The equations do not have classical solutions and the PI wants to extend the theory of so called L^p-viscosity solutions to this class. In particular the PI plans to investigate the question of regularity of solutions of such equations. This is a major open problem of elliptic partial differential equations and the PI proposes several possible new approaches to it that may give rise to new and interesting techniques.The notion of viscosity solution is one of the main tools of nonlinear partial differential equations and it has found applications in areas as diverse as optimal control, image processing, moving fronts and phase transitions, statistical mechanics, economics, mathematical finance. The motivation for studying some problems described in the proposal comes from optimal control,especially control of stochastic partial differential equations. Their theory is in a state of rapid development and is fueled by modeling questions coming from physics, population biology, chemistry, and economics and mathematical finance. The problem of optimal control of fluid flow is one of the basic engineering problems and has numerous applications in areas like combustion theory, aero and hydrodynamic control, Tokomak magnetic fusion, ocean and atmospheric prediction just to name a few. Problems related to the Musiela model of interest rates come from the modern theory of option pricing. The research of the project should contribute to the development of new directions in partial differential equations and should also have impact on the applied areas mentioned above.