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.
Cite
@article{arxiv.2206.12537,
title = {Hamilton-Jacobi equations with monotone nonlinearities on convex cones},
author = {Hong-Bin Chen and Jiaming Xia},
journal= {arXiv preprint arXiv:2206.12537},
year = {2024}
}
Comments
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 $\Gamma^\nearrow_\cdots$ instead of $\Gamma_\cdots$; Nevertheless, proofs need no change; Also, an appendix is added to contain the alternative versions of these results when $\mathsf{F}$ is not monotone