Homogenization of semi-linear PDEs with discontinuous effective coefficients

2008/03/25 by Khaled Bahlali, Bahlali, K., Abouo Elouaflin +3
Computer Science · Economics, Econometrics and Finance · Mathematics · #35K60 #60H20 #60H30 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.0803.3499

openalex publication_date 2008/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the asymptotic behavior of solution of semi-linear PDEs. Neither periodicity nor ergodicity will be assumed. In return, we assume that the coefficients admit a limit in Cesaro sense. In such a case, the averaged coefficients could be discontinuous. We use probabilistic approach based on weak convergence for the associated backward stochastic differential equation in the 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 "L^p-viscosity solution" introduced in \citeCCKS. We use BSDEs techniques to establish the existence of L^p-viscosity solution for the averaged PDE. We establish weak continuity for the flow of the limit diffusion process and related the PDE limit to the backward stochastic differential equation via the representation of L^p-viscosity solution.

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