Some inverse problems in periodic homogenization of Hamilton-Jacobi equations

TL;DR

This paper investigates the inverse problem of determining the potential function V(x) from the effective Hamiltonian in periodic homogenization of Hamilton-Jacobi equations, exploring specific cases in convex and nonconvex settings.

Contribution

It formulates and analyzes inverse problems relating V and the effective Hamiltonian, providing insights into their relationship in various convex and nonconvex scenarios.

Findings

Identified conditions under which V can be recovered from the effective Hamiltonian.

Analyzed special cases in convex and nonconvex settings.

Discussed the challenges and potential approaches for inverse problems in homogenization.

Abstract

We look at the effective Hamiltonian $\bar H$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\bar H$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. In the paper, we discuss several special cases in both convex and nonconvex settings.