Quantitative homogenization of convex Hamilton-Jacobi equations with Neumann type boundary conditions

We study the periodic homogenization for convex Hamilton-Jacobi equations on perforated domains under the Neumann type boundary conditions. We consider two types of conditions, the oblique derivative boundary condition and the prescribed contact angle boundary condition, which is important in the front propagation. We first establish a new representation formula for the solution by using the Skorokhod problem and modified Lagrangians. By using this formula essentially, we prove the sub and superadditivity properties of the extended metric functions, which will be applied to obtain the optimal convergence rate $O(\varepsilon)$ for homogenization of Neumann type problems.

💡 Research Summary

This paper addresses the periodic homogenization of convex Hamilton‑Jacobi equations posed on perforated domains Ω_ε = εΩ, where Ω ⊂ ℝⁿ (n ≥ 2) is a connected open set with a C¹ periodic boundary. The authors consider two Neumann‑type boundary conditions: an oblique derivative condition B_N(y,p) = γ(y)·p – g(y) with a continuous vector field γ satisfying γ·ν > 0 on ∂Ω, and a prescribed contact‑angle condition B_C(y,p) = ν(y)·p – cosθ(y) |p| where the contact angle θ(y) ∈