Min-max formulas and other properties of certain classes of nonconvex effective Hamiltonians

TL;DR

This paper introduces a new decomposition method to analyze the properties of effective Hamiltonians in nonconvex Hamilton-Jacobi equations, including min-max formulas, phenomena like quasi-convexification, and applications to stochastic homogenization.

Contribution

It presents a novel and robust decomposition technique for deriving min-max formulas for nonconvex effective Hamiltonians, advancing understanding of their properties and applications.

Findings

Derived min-max formulas for nonconvex Hamiltonians

Identified phenomena such as quasi-convexification and symmetry breakdown

Extended methods to stochastic homogenization of nonconvex equations

Abstract

This paper is the first attempt to systematically study properties of the effective Hamiltonian $\overline{H}$ arising in the periodic homogenization of some coercive but nonconvex Hamilton-Jacobi equations. Firstly, we introduce a new and robust decomposition method to obtain min-max formulas for a class of nonconvex $\overline{H}$. Secondly, we analytically and numerically investigate other related interesting phenomena, such as "quasi-convexification" and breakdown of symmetry, of $\overline{H}$ from other typical nonconvex Hamiltonians. Finally, in the appendix, we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for same class of nonconvex Hamilton-Jacobi equations. Some conjectures and problems are also proposed.