On cell problems for Hamilton-Jacobi equations with non-coercive Hamiltonians and its application to homogenization problems

TL;DR

This paper investigates cell problems in homogenization for Hamilton-Jacobi equations with non-coercive Hamiltonians, introducing generalized effective Hamiltonians and analyzing conditions for homogenization success or failure.

Contribution

It introduces a generalized notion of effective Hamiltonians for non-coercive Hamiltonians and characterizes cell problem solvability in this context.

Findings

Generalized effective Hamiltonians are characterized for non-coercive cases.

Homogenization can fail for non-coercive Hamilton-Jacobi equations.

Conditions for successful homogenization are identified.

Abstract

We study a cell problem arising in homogenization for a Hamilton-Jacobi equation whose Hamiltonian is not coercive. We introduce a generalized notion of effective Hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective Hamiltonian. Under some sufficient conditions, the result is applied to the associated homogenization problem. We also show that homogenization for non-coercive equations fails in general.