Representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians H. IshiiHiroyoshi Mitake 35F3035C9935F20representation formulaHamilton-Jacobi equationsAubry setsweak KAM theorystate constraint problem 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]. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3048 10.1512/iumj.2007.56.3048 en Indiana Univ. Math. J. 56 (2007) 2159 - 2184 state-of-the-art mathematics http://iumj.org/access/