Hamilton-Jacobi equations with monotone nonlinearities on convex cones

Abstract

We study the Cauchy problem of a Hamilton-Jacobi equation with the spatial variable in a closed convex cone. A monotonicity assumption on the nonlinearity allows us to prescribe no condition on the boundary of the cone. We show the well-posedness of the equation in the viscosity sense and prove several properties of the solution: monotonicity, Lipschitzness, and representations by variational formulas.

35 pages; Fixed an omission of a condition in the comparison principle, Proposition 3.1, which propagates to Corollary 3.2 and Proposition 4.1; More precisely, $\mathsf{F}$ should be in $\hat{I}{“}_{\cdots }^{\nearrow }$ instead of $\hat{I}{“}_{\cdots }$; Nevertheless, proofs need no change; Also, an appendix is added to contain the alternative versions of these results when $\mathsf{F}$ is not monotone