Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equations
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911100871639040 |
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| author | Han, Yuxi Tu, Son N. T. |
| author_facet | Han, Yuxi Tu, Son N. T. |
| contents | We investigate the convergence rate in the vanishing viscosity process of the solutions to the subquadratic state-constraint Hamilton-Jacobi equations. We give two different proofs of the fact that, for nonnegative Lipschitz data that vanish on the boundary, the rate of convergence is $\mathcal{O}(\sqrt{\varepsilon})$ in the interior. Moreover, the one-sided rate can be improved to $\mathcal{O}(\varepsilon)$ for nonnegative compactly supported data and $\mathcal{O}(\varepsilon^{1/(p-\frac{1}{2})})$ (where $1<p\leq 2$ is the exponent of the gradient term) for nonnegative data $f\in \mathrm{C}^2(\overlineΩ)$ such that $f = 0$ and $Df = 0$ on the boundary. Our approach relies on deep understanding of the blow-up behavior near the boundary and semiconcavity of the solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_09860 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equations Han, Yuxi Tu, Son N. T. Analysis of PDEs 35B40, 35D40, 49J20, 49L25, 70H20 We investigate the convergence rate in the vanishing viscosity process of the solutions to the subquadratic state-constraint Hamilton-Jacobi equations. We give two different proofs of the fact that, for nonnegative Lipschitz data that vanish on the boundary, the rate of convergence is $\mathcal{O}(\sqrt{\varepsilon})$ in the interior. Moreover, the one-sided rate can be improved to $\mathcal{O}(\varepsilon)$ for nonnegative compactly supported data and $\mathcal{O}(\varepsilon^{1/(p-\frac{1}{2})})$ (where $1<p\leq 2$ is the exponent of the gradient term) for nonnegative data $f\in \mathrm{C}^2(\overlineΩ)$ such that $f = 0$ and $Df = 0$ on the boundary. Our approach relies on deep understanding of the blow-up behavior near the boundary and semiconcavity of the solutions. |
| title | Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equations |
| topic | Analysis of PDEs 35B40, 35D40, 49J20, 49L25, 70H20 |
| url | https://arxiv.org/abs/2107.09860 |