Homogenization for nonlinear elliptic equations with random highly oscillatory coefficients. (English) Zbl 0686.60056

Partial differential equations and the calculus of variations. Essays in Honor of Ennio De Giorgi, 93-133 (1989).
[For the entire collection see Zbl 0671.00007.]
Using the theory of homogenization it is shown that if \(u^{\epsilon}\) is the solution of the nonlinear stochastic elliptic equation \[ (- \partial /\partial x_ i)a^{\epsilon}_{ij}(x;\omega)(\partial u^{\epsilon}/\partial x_ j)=H^{\epsilon}(x,Du^{\epsilon},u^{\epsilon},\omega),\quad u^{\epsilon}|_{\partial_ 0}=0 \] then \(u^{\epsilon}\to u\) strongly in \(L^ 2(\Omega,A,P;H^ 1_ 0(0))\) weakly and \(L^ 2(\Omega,A,P;L^ 2(0))\) strongly where (\(\Omega\),A,P) is a probability space and u is the solution of the equation \[ (-\partial /\partial x_ i)(q_{ij}\partial /\partial x_ j)u=H(x,Du,u,\omega), \] where \(H^{\epsilon}\to H\) in an appropriate sense and \(a^{\epsilon}_{ij}(x;\omega)\partial u^{\epsilon}/\partial x_ j\to q_{ij}\partial u/\partial x_ j.\)
Reviewer: S.P.Banks


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35J99 Elliptic equations and elliptic systems


Zbl 0671.00007