Title: Min-max formulas and other properties of certain classes of nonconvex effective Hamiltonians,
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.