## Electronic Journal of Probability

### Homogenization of semilinear PDEs with discontinuous averaged coefficients

#### Abstract

We study the asymptotic behavior of solutions of semilinear PDEs. Neither periodicity nor ergodicity will be assumed. On the other hand, we assume that the coecients have averages in the Cesaro sense. In such a case, the averaged coecients could be discontinuous. We use a probabilistic approach based on weak convergence of the associated backward stochastic dierential equation (BSDE) in the Jakubowski $S$-topology to derive the averaged PDE. However, since the averaged coecients are discontinuous, the classical viscosity solution is not dened for the averaged PDE. We then use the notion of "$L_p$-viscosity solution" introduced in [7]. The existence of $L_p$-viscosity solution to the averaged PDE is proved here by using BSDEs techniques.

#### Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 18, 477-499.

Dates
Accepted: 22 February 2009
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819479

Digital Object Identifier
doi:10.1214/EJP.v14-627

Mathematical Reviews number (MathSciNet)
MR2480550

Zentralblatt MATH identifier
1190.60055

Rights

#### Citation

Bahlali, Khaled; Elouaflin, A; Pardoux, Etienne. Homogenization of semilinear PDEs with discontinuous averaged coefficients. Electron. J. Probab. 14 (2009), paper no. 18, 477--499. doi:10.1214/EJP.v14-627. https://projecteuclid.org/euclid.ejp/1464819479

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