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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 the same class of nonconvex Hamilton–Jacobi equations. Some conjectures and problems are also proposed.

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