Quantitative homogenization for static contact Hamilton-Jacobi equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914498700378112 |
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| author | Liu, Gengyu Tu, Son N. T. Zhang, Jianlu |
| author_facet | Liu, Gengyu Tu, Son N. T. Zhang, Jianlu |
| contents | We characterize possible pairs $(u_\varepsilon,c)\in C(\mathbb{R}^n\backslash\varepsilon\mathbb{Z}^n,\mathbb{R})\times\mathbb{R}$ addressing the homogenization problem for Hamilton--Jacobi equations $$
H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=c,
\quad
\left({\mathrm resp.} \quad
H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=\varepsilonΔu_\varepsilon+c \right) $$ for all $\varepsilon>0$. Under a (not necessarily strict) monotonicity assumption on the Hamiltonian, we proposed certain criteria (based on the structure of Mather measures), under which all possible solutions $u_\varepsilon$ converge to a uniquely identified limit $u\in C(\mathbb{R}^n,\mathbb{R})$ solving the effective equation \[ \overline H( du,u)=c,\quad ({\mathrm resp.}\quad \overline H(du,u)=Δu+c) \] as $\varepsilon\rightarrow 0_+$ with a uniform rate $\mathcal{O}(\varepsilon)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02693 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative homogenization for static contact Hamilton-Jacobi equations Liu, Gengyu Tu, Son N. T. Zhang, Jianlu Analysis of PDEs 35B27, 35D40, 37K99, 49L25, 70H20, 74Q10, 78M40 We characterize possible pairs $(u_\varepsilon,c)\in C(\mathbb{R}^n\backslash\varepsilon\mathbb{Z}^n,\mathbb{R})\times\mathbb{R}$ addressing the homogenization problem for Hamilton--Jacobi equations $$ H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=c, \quad \left({\mathrm resp.} \quad H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=\varepsilonΔu_\varepsilon+c \right) $$ for all $\varepsilon>0$. Under a (not necessarily strict) monotonicity assumption on the Hamiltonian, we proposed certain criteria (based on the structure of Mather measures), under which all possible solutions $u_\varepsilon$ converge to a uniquely identified limit $u\in C(\mathbb{R}^n,\mathbb{R})$ solving the effective equation \[ \overline H( du,u)=c,\quad ({\mathrm resp.}\quad \overline H(du,u)=Δu+c) \] as $\varepsilon\rightarrow 0_+$ with a uniform rate $\mathcal{O}(\varepsilon)$. |
| title | Quantitative homogenization for static contact Hamilton-Jacobi equations |
| topic | Analysis of PDEs 35B27, 35D40, 37K99, 49L25, 70H20, 74Q10, 78M40 |
| url | https://arxiv.org/abs/2604.02693 |