Viscosity Solutions of Fully Nonlinear Equations
CALIFORNIA UNIV SANTA BARBARA
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The eight publications produced by the project established a number of basic results in the theory of viscosity solutions of fully nonlinear differential equations of first and second order in finite and infinite dimensions. These equations arise in the dynamic programming theory of control and differential games the finite dimensional theory for ode and the infinite dimensional theory for pde dynamics. Being fully nonlinear, the equations do not typically admit regular or classical solutions, and the appropriate notion is that of viscosity solutions. Two major advances in the first order infinite dimensional case consisted of determining the precise notion appropriate to a class of infinite dimensional problems with unbounded terms arising from the pde dynamics, and the examination of a limit case in which the value function is not a solution, but the maximal subsolution. Significant contributions to the second order theory include a new exposition of the finite dimensional theory based on results from previous funding, an infinite dimensional generalization of the foundational result used in this exposition, and the extension of the theory to second order equations in infinite dimensions with unbounded first order terms.
- Numerical Mathematics