Periodic Homogenization of Green and Neumann Functions

Abstract

For a family of second-order elliptic operators with rapidly oscillating periodic coefficients, we study the asymptotic behavior of the Green and Neumann functions, using Dirichlet and Neumann correctors. As a result we obtain asymptotic expansions of Poisson kernels and the Dirichlet-to-Neumann maps as well as optimal convergence rates in L p and W 1;p for solutions with Dirichlet or Neumann boundary conditions.

Boundary Layers in Periodic Homogenization of Neumann Problems

“…The estimate (1.13) was proved in [19] for bounded C 2;˛d omains, using boundary Lipschitz estimates for solutions with Neumann data [18] (also see [6], which extends the estimates to operators without the symmetry condition A D A). y/g @ @y`f N 0 .x; y/gˇÄ C " 1 jx yj d for any x; y 2 and 2 .0; 1/.…”

“…In fact, it was proved in [19] that if is a bounded C 1;1 domain in R d and d 3, then (1.12) jN " .x; y/ N 0 .x; y/j Ä C " lnOE" 1 jx yj C 2 jx yj d 2 In fact, it was proved in [19] that if is a bounded C 1;1 domain in R d and d 3, then (1.12) jN " .x; y/ N 0 .x; y/j Ä C " lnOE" 1 jx yj C 2 jx yj d 2…”

Commutator estimates for the Dirichlet-to-Neumann map of Stokes systems in Lipschitz domains

“…We hope our results may be further applied to the study of fluid mechanics. For more knowledges on this subject, we refer the readers to [1,3,5,6,9,14,18,15,16] for more details and references therein.…”

Convergence Rates for the Stratified Periodic Homogenization Problems

“…[15] have also studied the asymptotic behavior of the Green and Neumann functions and obtained some error estimates for solutions.…”

“…In this paper, we overcome this problem after introduced auxiliary function. The procedure we used for obtaining convergence rates estimates is somewhat analogous to the process Kenig, Lin and Shen [15] used for the most classical homogenization problems. The main purpose of this paper is to extend their [15] …”