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 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 H. Secondly, we analytically and numerically investigate other related interesting phenomena, such as "quasi-convexification" and breakdown of symmetry, of 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.2010 Mathematics Subject Classification. 35B10 35B20 35B27 35D40 35F21 .

Stochastic homogenization of certain nonconvex Hamilton–Jacobi equations

“…Instead, one considers a convergence property, i.e., the regularly homogenizability (see Definition 2), of the auxiliary macroscopic problem (2.1). This has been established for the aforementioned Hamiltonians (see [28,4,1,6,7,18,30], etc.). Recently, Cardaliaguet and Souganidis [8] proved the existence of the cell problem for all extreme points of the convex hull of the sublevel sets of the effective Hamiltonian, if it exists.…”

“…This result was extended by Feldman and Souganidis [16] to Hamiltonians with star-shaped sub-level sets. Recently, Qian, Tran and Yu [30] provided a new decomposition method to prove homogenization for some general classes of even separable nonconvex Hamiltonians in multi dimensions, which include the result in [6] as a special case. In the case of second-order nonconvex Hamilton-Jacobi equations, the homogenization result was proved if d = 1 and the Hamiltonian takes certain special forms.…”

Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension

“…The effective Hamiltonian H(x, p) : R 2n −→ R is determined by H in a very nonlinear way through the cell problem as follows: For each (x, p) ∈ R n × R n , it can be shown (see [12] and [8,9]) that there is a unique constant λ = λ(x, p) ∈ R for which the following cell problem It is worth mentioning that in general v(y; x, p) is not unique even up to adding a constant. More research in understanding the effective Hamiltonians H is reported in [5,6,13,18] and the references therein.…”

Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential

“…For quite some time it was not clear whether the convexity assumption can be disposed of in the stochastic setting. Several classes of examples of nonconvex Hamiltonians for which homogenization holds were recently constructed: [ATY15], [ATY16], [Gao16], [FS16], [QTY17] for inviscid equations and [AC15b], [DK17] for viscous equations. On the other hand, the work [Zil15] has demonstrated that homogenization could fail for nonconvex Hamiltonians in the general stationary and ergodic setting for dimensions d ≥ 2.…”