The averaging of nondivergence second order elliptic and parabolic operators and the stabilization of solutions of the Cauchy problem

V. V. Zhikov ; M. M. Sirazhudinov

Sbornik. Mathematics, Tome 44 (1983) no. 2, pp. 149-166

Résumé

Let $\{a_{ij}(x)\}$ ($i,j=1,\dots,n$) be an elliptic matrix, where the $a_{ij}(x)$ are almost periodic functions in the sense of Bohr. In the case $n\geqslant3$ it is assumed that Bernstein's inequality holds. Problems of averaging families of elliptic $A_\varepsilon=a_{ij}(\varepsilon^{-1}x)D_iD_j$ and parabolic $L_\varepsilon=\frac\partial{\partial t}-a_{ij}(\varepsilon^{-1}x)D_iD_j$ operators are considered, and a criterion for pointwise and uniform stabilization is obtained for the solution of the Cauchy problem. A key role in these questions is played by a nonnegative solution of the equation $A^*p=D_iD_j(a_{ij}p)=0$. In particular it is proved that the equation has a unique (up to a multiplicative factor) solution in a class of almost periodic functions in the sense of Besicovitch. A stronger ergodic theorem (or uniqueness of “'stationary distribution”) is also proved: the equation $A^*f=0$ has a unique (up to a multiplicative factor) solution in the dual of the space of Bohr almost periodic functions. The case of periodic coefficients is also considered (when the equation is parabolic it is assumed to be time dependent), and averaging and stabilization theorems without Bernstein's inequality are proved. Bibliography: 17 titles.