Journal
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
Volume 108, Issue 5, Pages 751-782Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.matpur.2017.05.016
Keywords
Stochastic homogenization; Hamilton-Jacobi equations; Viscosity solutions
Categories
Funding
- NSF-RTG grant [DMS-1246999]
- NSF grants [DMS-1266383, DMS-1600129]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1266383] Funding Source: National Science Foundation
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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 counterexample of Ziliotto, 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 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. (C) 2017 Elsevier Masson SAS. All rights reserved.
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