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arXiv · 2509.01973

Rate of convergence of the vanishing viscosity method for Hamilton-Jacobi equations with Neumann boundary conditions

Alessandro Goffi
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Abstract

We study the quantitative small noise limit in the $L^\infty$ norm of certain time-dependent Hamilton-Jacobi equations equipped with Neumann boundary conditions, depending on the regularity of the data and the geometric properties of the domain. We first provide a $\mathcal{O}(\sqrt{\varepsilon})$ rate of convergence for Hamilton-Jacobi equations with locally Lipschitz Hamiltonians posed on convex domains of the Euclidean space. We then enhance this speed of convergence in the case of quadratic Hamiltonians proving one-side rates of order $\mathcal{O}(\varepsilon)$ and $\mathcal{O}(\varepsilon^\beta)$, $\beta\in(1/2,1)$. The results exploit recent $L^1$ contraction estimates for Fokker-Planck equations with bounded velocity fields on unbounded domains used to derive differential Harnack estimates for the corresponding Neumann heat flow.

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Alessandro Goffi. 2025-09-02. Rate of convergence of the vanishing viscosity method for Hamilton-Jacobi equations with Neumann boundary conditions. https://arxiv.org/abs/2509.01973

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