Viscosity Solutions of Minimization Problems

It is proved, in a rather large setting, that the solutions of the approximate problems converge to a ``viscosity solution'' of the original problem, that is, a solution that is minimal among all the solutions with respect to some viscosity criteria.

Abstract

Viscosity methods for minimization problems are revisited from some modern perspectives in variational analysis. Variational convergences for sequences of functions (epi-convergence, $\Gamma $-convergence, Mosco-convergence) and for sequences of operators (graph-convergence) provide a flexible tool for such questions. It is proved, in a rather large setting, that the solutions of the approximate problems converge to a "viscosity solution" of the original problem, that is, a solution that is minimal among all the solutions with respect to some viscosity criteria. Various examples coming from mathematical programming, calculus of variations, semicoercive elliptic equations, phase transition theory, Hamilton–Jacobi equations, singular perturbations, and optimal control theory are considered.