Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result

The main result of the paper is a general convergence theorem for the viscosity solutions of singular perturbation problems for fully nonlinear degenerate parabolic PDEs (partial differential equations) with highly oscillating initial data. It substantially generalizes some results obtained previously in [2]. Under the only assumptions that the Hamiltonian is ergodic and stabilizing in a suitable sense, the solutions are proved to converge in a relaxed sense to the solution of a limit Cauchy problem with appropriate effective Hamiltonian and initial data. In its formulation, our convergence result is analogous to the stability property of Barles and Perthame. It should thus reveal a useful tool for studying general singular perturbation problems by viscosity solutions techniques. A detailed exposition of ergodicity and stabilization is given, with many examples. Applications to homogenization and averaging are also discussed.