VISCOSITY SOLUTIONS OF MINIMIZATION PROBLEMS

Viscosity methods for minimization problems are revisited-from some mo dern perspectives in variational analysis. Variational convergences fo r sequences of functions (epi-convergence, Gamma-convergence, Mosco-co nvergence) and for sequences of operators (graph-convergence) provide a flexible tool for such questions. It is proved, in a rather large se tting, 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 cri teria. Various examples coming from mathematical programming, calculus of variations, semicoercive elliptic equations, phase transition theo ry, Hamilton-Jacobi equations, singular perturbations, and optimal con trol theory are considered.