Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 333, Issue -, Pages 231-267Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2022.06.011
Keywords
Homogenization; Equations in media with random structure; Nonconvex Hamiltonian; Viscous Hamilton-Jacobi equation; Random potential
Categories
Funding
- Simons Foundation [523625]
- Sapienza Universita di Roma
- Fields Institute for Research in Mathematical Sciences
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We prove homogenization for a class of nonconvex (possibly degenerate) viscous Hamilton-Jacobi equations in stationary ergodic random environments in one space dimension. The results concern Hamiltonians of the form G(p) + V (x, omega), where the nonlinearity G is the minimum of two or more convex functions with the same absolute minimum, and the potential V is a bounded stationary process satisfying an additional scaled hill and valley condition. This condition is trivially satisfied in the inviscid case, while it is equivalent to the original hill and valley condition of A. Yilmaz and O. Zeitouni [32] in the uniformly elliptic case. Our approach is based on PDE methods and does not rely on representation formulas for solutions. Using only comparison with suitably constructed super- and sub- solutions, we obtain tight upper and lower bounds for solutions with linear initial data x ? theta x. Another important ingredient is a general result of P. Cardaliaguet and P. E. Souganidis [13] which guarantees the existence of sublinear correctors for all theta outside flat parts of effective Hamiltonians associated with the convex functions from which G is built. We derive crucial derivative estimates for these correctors which allow us to use them as correctors for G. (c) 2022 Elsevier Inc. All rights reserved.
We prove homogenization for a class of nonconvex (possibly degenerate) viscous Hamilton-Jacobi equations in stationary ergodic random environments in one space dimension. The results concern Hamiltonians of the form G(p) + V (x, omega), where the nonlinearity G is the minimum of two or more convex functions with the same absolute minimum, and the potential V is a bounded stationary process satisfying an additional scaled hill and valley condition. This condition is trivially satisfied in the inviscid case, while it is equivalent to the original hill and valley condition of A. Yilmaz and O. Zeitouni [32] in the uniformly elliptic case. Our approach is based on PDE methods and does not rely on representation formulas for solutions. Using only comparison with suitably constructed super- and sub- solutions, we obtain tight upper and lower bounds for solutions with linear initial data x ? theta x. Another important ingredient is a general result of P. Cardaliaguet and P. E. Souganidis [13] which guarantees the existence of sublinear correctors for all theta outside flat parts of effective Hamiltonians associated with the convex functions from which G is built. We derive crucial derivative estimates for these correctors which allow us to use them as correctors for G.(c) 2022 Elsevier Inc. All rights reserved.
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