Lower gradient estimates for viscosity solutions to first-order Hamilton--Jacobi equations depending on the unknown function
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911945665282048 |
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| author | Hirose, Kazuya |
| author_facet | Hirose, Kazuya |
| contents | In this paper, we derive the lower bounds for the gradients of viscosity solutions to the Hamilton--Jacobi equation, where the convex Hamiltonian depends on the unknown function. We obtain gradient estimates using two different methods. First, we utilize the equivalence between viscosity solutions and Barron--Jensen solutions to study the properties of the inf-convolution. Second, we examine the Lie equation to understand how initial gradients propagate along its solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04288 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower gradient estimates for viscosity solutions to first-order Hamilton--Jacobi equations depending on the unknown function Hirose, Kazuya Analysis of PDEs In this paper, we derive the lower bounds for the gradients of viscosity solutions to the Hamilton--Jacobi equation, where the convex Hamiltonian depends on the unknown function. We obtain gradient estimates using two different methods. First, we utilize the equivalence between viscosity solutions and Barron--Jensen solutions to study the properties of the inf-convolution. Second, we examine the Lie equation to understand how initial gradients propagate along its solutions. |
| title | Lower gradient estimates for viscosity solutions to first-order Hamilton--Jacobi equations depending on the unknown function |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.04288 |