Abstract:
For the equation with rapidly oscillating coefficients in a bounded domain $\mathscr O\subset\mathbf R^n$ $$
\sum_{i,j=1}^n\frac\partial{\partial x_i}a_{ij}\biggl(\frac x\varepsilon\biggr),\qquad u_\varepsilon(x)|_{\partial\mathscr O}=f_1(x),
$$
where $(a_{ij}(y))$ form homogeneous random fields, an averaged equation of the form
$$
\sum_{i,j=1}^nq_{ij}\frac{\partial^2} {\partial x_i\partial x_j}u_0(x)=f(x),\qquad
u_0(x)|_{\partial\mathscr O}=f_1(x),
$$
is constructed with coefficients $q_{ij}$ which do not depend on $x$; various applications of this result are also obtained.
Bibliography: 22 titles.
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