Elliptic perturbations for a class of Hamilton–Jacobi equations

Abstract

A Hamilton–Jacobi equation, with a convex Hamiltonian such that the corresponding Langrangean vanishes along the orbits of certain dynamical systems, is considered. Small elliptic and inhomogeneous perturbations of such HJE are studied in bounded domains. In the problem at hand the HJE may have more than one viscosity solution under zero boundary conditions, which brings up the question of uniqueness conditions. It is found that under the uniqueness conditions the problem can be reduced to the linear case by a completely new method (notice that the logarithmic transformation, generally speaking, cannot be applied in the considered problem).