A note on the stochastic weakly* almost periodic homogenization of fully nonlinear elliptic equations

  • Hermano Frid

    Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil


A function fBUC(Rd)f\in \mathrm {BUC}(\mathbb R^d) is said to be weakly* almost periodic, denoted fWAP(Rd)f\in \mathcal W^* \mathrm {AP}(\mathbb R^d), if there is gAP(Rd)g \in \mathrm {AP}(\mathbb R^d), such that, M(fg)=0\mathrm {M}(|f-g|)=0, where BUC(Rd)\mathrm {BUC}(\mathbb R^d) and AP(Rd)\mathrm {AP}(\mathbb R^d) are, respectively, the space of bounded uniformly continuous functions and the space of almost periodic functions, in Rd\mathbb R^d, and M(h)\mathrm M(h) denotes the mean value of hh, if it exists. We give a very simple direct proof of the stochastic homogenization property of the Dirichlet problem for fully nonlinear uniformly elliptic equations of the form F(ω,xε,D2u)=0F(\omega,\frac{x}{\varepsilon},D^2u)=0, xUx \in U, in a bounded domain URdU\subset \mathbb R^d, in the case where for almost all ωΩ\omega \in \Omega, the realization F(ω,,M)F(\omega,\cdot, M) is a weakly* almost periodic function, for all MSdM \in \mathcal S^d, where Sd\mathcal S^d is the space of d×dd \times d symmetric matrices. Here (Ω,μ,F)(\Omega,\mu,\mathcal F) is a probability space with probability measure μ\mu and σ\sigma-algebra F\mathcal F of μ\mu-measurable subsets of Ω\Omega. For each fixed MSdM \in \mathcal S^d, F(ω,y,M)F(\omega, y, M) is a stationary process, that is, F(ω,y,M)=F~(T(y)ω,M):=F(T(y)ω,0,M)F (\omega, y, M)=\tilde F(T (y) \omega, M):= F (T (y) \omega, 0, M), where T(y):ΩΩT(y):\Omega \to \Omega is an ergodic group of measure preserving mappings such that the mapping (ω,y)T(y)ω(\omega, y) \to T(y) \omega is measurable. Also, F(ω,y,M)F(\omega, y, M), MSdM \in \mathcal S^d, is uniformly elliptic, with ellipticity constants 0<λ<Λ0< \lambda <\Lambda independent of (ω,y)Ω×Rd(\omega, y) \in \Omega \times \mathbb R^d. The result presented here is a particular instance of the general theorem of Caffarelli, Souganidis and Wang, in CPAM 2005. Our point here is just to show a straightforward proof for this special case, which serves as a motivation for that general theorem, whose proof involves much more intricate arguments. We remark that any continuous stationary process verifies the property that almost all realizations belong to an ergodic algebra on Rd\mathbb R^d, and that WAP(Rd)\mathcal W ^* \mathrm {AP} (\mathbb R^d) contains all the ergodic algebras on Rd\mathbb R^d so far known.

Cite this article

Hermano Frid, A note on the stochastic weakly* almost periodic homogenization of fully nonlinear elliptic equations. Port. Math. 72 (2015), no. 2/3, pp. 207–227

DOI 10.4171/PM/1965