Bounded-from-below viscosity solutions of Hamilton-Jacobi equations

Abstract

We establish a uniqueness and existence theorem for bounded-from-below viscosity solutions of Hamilton-Jacobi equations of the form $u+H(Du)=f$ in $\mathbb{R}^N$. More precisely, we show that there is a unique solution $u$ such that $u^-$ grows at most linearly, when $f^-$ behaves analogously and when $H$ is convex and nonlinear. We sharpen this result when the behavior of the Hamiltonian is known at infinity. We also discuss some extensions to more general Hamiltonians and to the Dirichlet problem in an unbounded open set.