article

Title: Representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians

Authors: Hitoshi Ishii and Hiroyoshi Mitake

Issue: Volume 56 (2007), Issue 5, 2159-2184

Abstract:

$We establish general representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians. In order to treat representation formulas on general domains, we introduce a notion of ideal boundary similar to the Martin boundary [R.S. Martin, \textit{Minimal positive harmonic functions}, Trans. Amer. Math. Soc. \textbf{49} (1941), 137--172] in potential theory. We apply such representation formulas to investigate maximal solutions, in certain classes of functions, of Hamilton-Jacobi equations. Part of the results in this paper has been announced in [H. Mitake, \textit{A representation formula for solutions of the Hamilton-Jacobi equation. Viscosity solution theory of differential equations and its developments}, Surikaisekikenkyusho Kokyuroku \textbf{1481} (2006), 32--42].$