HOMOGENIZATION FOR NON-SELF-ADJOINT PERIODIC ELLIPTIC OPERATORS ON AN INFINITE CYLINDER

作者：Senik Nikita N

来源：SIAM Journal on Mathematical Analysis, 2017, 49(2): 874-898.

DOI：10.1137/15M1049981

• homogenization
• operator error estimates
• periodic differential operators
• effective operator
• corrector

We consider the problem of homogenization for non-self-adjoint second-order elliptic differential operators A(epsilon) of divergence form on L-2 (R-d1 x T-d2), where d(1) is positive and d(2) is non-negative. The coefficients of the operator A(epsilon) are periodic in the first variable with period epsilon and smooth in a certain sense in the second. We show that, as epsilon gets small, (A(epsilon) - mu)(-1) and del x(2) (A(epsilon) - mu)(-1) for an appropriate mu converge in the operator norm to, respectively, (A(0) - mu)(-1) and del x(2) (A(0) - mu)(-1), where A(0) is an operator whose coefficients depend only on x(2). We also obtain an approximation for del x(1) (A(epsilon) - mu)(-1) and find the next term in the approximation for (A(epsilon) - mu)(-1). Estimates for the rates of convergence and the rates of approximation are provided and are sharp with respect to the order.

• 出版日期2017