HOMOGENIZATION OF THE DIRICHLET PROBLEM FOR HIGHER-ORDER ELLIPTIC EQUATIONS WITH PERIODIC COEFFICIENTS

摘要

Let O subset of R-d be a bounded domain of class C-2p. The object under study is a selfadjoint strongly elliptic operator A(D,epsilon) of order 2p, p = 2, in L-2(O;C-n), given by the expression b(D)* g(x/epsilon)b(D), epsilon 0, with the Dirichlet boundary conditions. Here g(x) is a bounded and positive definite (m x m)-matrix-valued function in R-d, periodic with respect to some lattice; b(D) = Sigma(|alpha|=p) b(alpha)D(alpha) is a differential operator of order p with constant coefficients; and the b(alpha) are constant (m x n)-matrices. It is assumed that m = n and the symbol b(xi) has maximal rank. Approximations are found for the resolvent (Lambda(D,epsilon) -zeta I)(-1) in the L-2(O;C-n)-operator norm and in the norm of operators acting from L-2(O;C-n) to H-p(O;C-n), with error estimates depending on epsilon and zeta.