Quantitative homogenization of convex Hamilton-Jacobi equations with $u/\varepsilon$-periodic Hamiltonians
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| Format: | Preprint |
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2025
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| _version_ | 1866911031807180800 |
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| author | Mitake, Hiroyoshi Ni, Panrui Tran, Hung V. |
| author_facet | Mitake, Hiroyoshi Ni, Panrui Tran, Hung V. |
| contents | Here, we study quantitative homogenization of first-order convex Hamilton-Jacobi equations with $(u/\varepsilon)$-periodic Hamiltonians which typically appear in dislocation dynamics. Firstly, we establish the optimal convergence rate by using the inherent fundamental solution and the implicit variational principle of Hamilton dynamics with their Hamiltonian depending on the unknown. Secondly, under additional growth assumptions on the Hamiltonian, we establish global Hölder regularity for both the solutions and the correctors, serving as a notable application of our quantitative homogenization theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_00663 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative homogenization of convex Hamilton-Jacobi equations with $u/\varepsilon$-periodic Hamiltonians Mitake, Hiroyoshi Ni, Panrui Tran, Hung V. Analysis of PDEs 35B27, 35B40, 49L25 Here, we study quantitative homogenization of first-order convex Hamilton-Jacobi equations with $(u/\varepsilon)$-periodic Hamiltonians which typically appear in dislocation dynamics. Firstly, we establish the optimal convergence rate by using the inherent fundamental solution and the implicit variational principle of Hamilton dynamics with their Hamiltonian depending on the unknown. Secondly, under additional growth assumptions on the Hamiltonian, we establish global Hölder regularity for both the solutions and the correctors, serving as a notable application of our quantitative homogenization theory. |
| title | Quantitative homogenization of convex Hamilton-Jacobi equations with $u/\varepsilon$-periodic Hamiltonians |
| topic | Analysis of PDEs 35B27, 35B40, 49L25 |
| url | https://arxiv.org/abs/2507.00663 |