Homogenization and non-homogenization of certain non-convex Hamilton–Jacobi equations

Abstract

Abstract. We continue the study of the homogenization of coercive non-convex Hamilton-Jacobi equations in random media identifying two general classes of Hamiltonians with very distinct behavior. For the first class there is no homogenization in a particular environment while for the second homogenization takes place in environments with finite range dependence. Motivated by the recent counter-example of Ziliotto [18], who constructed a coercive but non-convex Hamilton-Jacobi equation with stationary ergodic random potential field for which homogenization does not hold, we show that same happens for coercive Hamiltonians which have a strict saddle-point, a very local property. We also identify, based on the recent work of Armstrong and Cardaliaguet [1] on the homogenization of positively homogeneous random Hamiltonians in environments with finite range dependence, a new general class Hamiltonians, namely equations with uniformly strictly star-shaped sub-level sets, which homogenize.

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

“…Ziliotto [42] gave a counterexample to homogenization of (4.1) in case n = 2. See also the paper by Feldman and Souganidis [21]. Basically, [42,21] implies that, in some specific cases, even if H has strict saddle points, H − V is still regularly homogenizable for every V with large oscillation.…”

Stochastic Homogenization for Functionals with Anisotropic Rescaling and Noncoercive Hamilton--Jacobi Equations

“…The case of non-convexity has been understood better recently, see e.g. [4,24,37]. Instead this paper gives a very general class of examples which are convex (but not strictly) but non-coercive.…”

On the existence of correctors for the stochastic homogenization of viscous Hamilton–Jacobi equations

“…With the exception of a case with Hamiltonians of a very special form (see Armstrong, Tran and Yu [3,5]), the main results known in nonconvex settings are quantitative. That is it is necessary to make some strong assumptions on the environment (finite range dependence) and to use sophisticated concentration inequalities to prove directly that the solutions of the oscillatory problems converge; see, for example, Armstrong and Cardaliaguet [3] and Feldman and Souganidis [8]. It should be noted that the counterexamples of Ziliotto [17] and [8] yield that in the setting of nonconvex homogenization in random media is not possible to prove the existence of correctors for all directions.…”