Homogenization of higher-order parabolic systems in a bounded domain
Tatiana Aleksandrovna Suslina · Applicable Analysis · 2017
Let O⊂Rd be a bounded domain of class C2p. In L2(O;Cn), we consider matrix elliptic differential operators AD,ε and AN,ε of order 2p (p⩾2) with the Dirichlet or Neumann boundary conditions, respectively. The coefficients of AD,ε and AN,ε are periodic and depend on x/ε, ε>0. The behavior of the operator e-A†,εt, †=D,N, for small ε is studied. It is shown that, for fixed t>0, the operator e-A†,εt converges in the L2-operator norm to e-A†0t, as ε→0. Here A†0 is the effective operator with constant coefficients. We obtain a sharp order estimate ‖e-A†,εt-e-A†0t‖L2→L2⩽Cε. Also, we find approximation for e-A†,εt in the (L2→Hp)-norm with error estimate of order O(ε1/2). The results are applied to homogenization of the solutions of initial boundary value problems for parabolic systems.