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arXiv · 2107.09860

Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equations

Yuxi Han·Son N. T. Tu
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Abstract

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.

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Yuxi Han, Son N. T. Tu. 2025-08-11. Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equations. https://doi.org/10.1007/s00245-022-09874-z

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