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Homogenization of semilinear PDEs with discontinuous averaged coefficients. (English) Zbl 1190.60055

Summary: We study the asymptotic behavior of solutions of semilinear PDEs. Neither periodicity nor ergodicity are assumed. On the other hand, we assume that the coefficients have averages in the Cesàro sense. In such a case, the averaged coefficients could be discontinuous. We use a probabilistic approach based on weak convergence of the associated backward stochastic differential equation (BSDE) in the Jakubowski \(S\)-topology to derive the averaged PDE. However, since the averaged coefficients are discontinuous, the classical viscosity solution is not defined for the averaged PDE. We then use the notion of “\(Lp\)-viscosity solution” introduced in [L. Caffarelli, M. G. Crandall, M. Kocan and A. Święch, Commun. Pure Appl. Math. 49, No. 4, 365–397 (1996; Zbl 0854.35032)]. The existence of \(Lp\)-viscosity solution to the averaged PDE is proved here by using BSDEs techniques.

MSC:

60H20 Stochastic integral equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations

Citations:

Zbl 0854.35032
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