Hamilton-Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown

Hitoshi Ishii; Kaizhi Wang; Lin Wang; Jun Yan

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arXiv · 2108.11216

Hamilton-Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown

Hitoshi Ishii·Kaizhi Wang·Lin Wang·Jun Yan

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Abstract

We study the Hamilton-Jacobi equations $H(x,Du,u)=0$ in $M$ and $\partial u/\partial t +H(x,D_xu,u)=0$ in $M\times(0,\infty)$, where the Hamiltonian $H=H(x,p,u)$ depends Lipschitz continuously on the variable $u$. In the framework of the semicontinuous viscosity solutions due to Barron-Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem.

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Hitoshi Ishii, Kaizhi Wang, Lin Wang, Jun Yan. 2021-08-25. Hamilton-Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown. https://arxiv.org/abs/2108.11216

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