Homogenization of higher-order parabolic systems in a bounded domain

摘要

Let O subset of R-d be a bounded domain of class C-2p. In L-2(O; C-n), we consider matrix elliptic differential operators A(D,epsilon) and A(N,epsilon) of order 2p (p = 2) with the Dirichlet or Neumann boundary conditions, respectively. The coefficients of A(D,epsilon) and A(N,epsilon) are periodic and depend on x/epsilon, epsilon 0. The behavior of the operator e(-A dagger,epsilon t,)dagger = D, N, for small epsilon is studied. It is shown that, for fixed t 0, the operator e(-A dagger,epsilon t) converges in the L-2-operator norm to e(-A dagger 0t), as epsilon - 0. Here A(dagger)(0) is the effective operator with constant coefficients. We obtain a sharp order estimate parallel to e(-A dagger,epsilon t)-e(-A dagger 0t) parallel to L-2 - L-2 = C epsilon. Also, we find approximation for e(-A dagger,epsilon t) in the (L-2 - H-p)-norm with error estimate of order O(epsilon(1/2)). The results are applied to homogenization of the solutions of initial boundary value problems for parabolic systems.