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

Hiroyoshi Mitake, Panrui Ni

Abstract

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.

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

Abstract

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 for homogenization of Neumann type problems.
Paper Structure (9 sections, 25 theorems, 232 equations, 7 figures)

This paper contains 9 sections, 25 theorems, 232 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Value function
  3. An example for the level set equation
  4. Basic properties of value functions
  5. Metric functions
  6. Definition of metric functions, and basic properties
  7. Difference of metric functions and extended metric functions
  8. Subadditivity and Superadditivity
  9. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Assume that (A1)--(A4) hold.

Figures (7)

  • Figure 1: Image picture of the optimal trajectory
  • Figure 2: Front propagation of $0$-level set of $u$
  • Figure 3: Image picture of the optimal trajectory
  • Figure 4: Front propagation of $0$-level set of $u$
  • Figure 5: Image Picture of $\eta_1$: (a) $\eta_1(a_1)$, (b) $\eta_1(b_1)$, (c) $\eta_1(a_2)$, (d) $\eta_1(b_2)$, (e) $\eta_1(a_3)$, (f) $\eta_1(b_3)$.
  • ...and 2 more figures

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1
  • Theorem 2.1: Comparison principle for \ref{['eq:CN1']}--\ref{['eq:CN2']}
  • Proposition 2.2: I11
  • Proposition 2.3: I11
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 37 more