HOMOGENIZATION OF INITIAL BOUNDARY VALUE PROBLEMS FOR PARABOLIC SYSTEMS
Tatiana Aleksandrovna Suslina · arXiv (Cornell University) · 2015
Let OR d be a bounded domain of class C 1,1 . In the Hilbert space L2(O;C n ), we consider matrix elliptic second order differ- ential operators AD,e and AN,e with the Dirichlet or Neumann boundary condition on @O, respectively. Here > 0 is the small parameter. The coefficients of the operators are periodic and depend onx/. The behav- ior of the operator e −A†,t , † = D,N, for small is studied. It is shown that, for fixed t > 0, the operator e −A†,t converges in the L2-operator norm to e −A 0 †t , as ! 0. Here A 0 † is the effective operator with con- stant coefficients. For the norm of the difference of the operators e −A †,t and e −A 0 †t a sharp order estimate (of order O()) is obtained. Also, we find approximation for the exponential e −A†,t in the (L2 ! H 1 )-norm with error estimate of order O( 1/2 ); in this approximation, a correc- tor is taken into account. The results are applied to homogenization of solutions of initial boundary value problems for parabolic systems.