Viscosity Solutions of Minimization Problems

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.