Stochastic homogenization of certain nonconvex Hamilton–Jacobi equations

Abstract

In this paper, we prove the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of quasiconvex Hamiltonians and a sequence of quasiconcave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenzation.2010 Mathematics Subject Classification. 35B27.

Stochastic homogenization and effective Hamiltonians of HJ equations in one space dimension: The double-well case

“…For other (positive and negative) results on the homogenization of HJ equations with nonconvex Hamiltonians, see [14,16,26].…”

An example of failure of stochastic homogenization for viscous Hamilton-Jacobi equations without convexity

“…This was proven in a series of works started by Armstrong, Tran and Yu [4,5] (for Hamiltonians in separated form) and completed by Gao [9] (general coercive Hamiltonians in d = 1). These techniques generalize to some higher dimensional problems with a special structure, see Gao [10].…”

Stochastic homogenization of a class of quasiconvex viscous Hamilton-Jacobi equations in one space dimension

“…If quasiconvexity is violated, then there is no general answer to the question of homogenization. Indeed, when d ≥ 2, there are positive results for certain classes of such HJ equations (see [4,6,1,25,18]) as well as negative results for others (see [30,15,14]). The counterexamples in the latter collection of papers involve Hamiltonians with saddle points, so they cannot be adapted to d = 1.…”