Remarks on the Existence and Uniqueness of Unbounded Viscosity Solutions of Hamilton-Jacobi Equations

摘要

This paper is concerned with various questions about the existence and uniqueness of solutions of Hamilton-Jacobi equations in RN. The issues treated have to do with the interaction between structure properties of the Hamiltonian (in particular, continuity and growth properties), properties of the solutions and the existence and uniqueness. Uniqueness is exhibited in appropriate growth classes depending on the Hamiltonian and existence is exhibited in these classes when the assumptions are slightly strengthened. Existence results are also given under assumptions for which uniqueness fails, existence of minimal solutions is shown given the existence of a subsolution, and examples are given to indicate the sharpness of some of the results.