$L^2$ estimates for commutators of the Dirichlet-to-Neumann Map associated to elliptic operators with complex-valued bounded measurable coefficients on ${\mathbb{R}}_{+}^{n+1}$

Abstract

In this paper we establish commmutator estimates for the Dirichlet-to-Neumann Map associated to a divergence form elliptic operator in the upper half-space ${\mathbb{R}}_{+}^{n+1}:=\{(x,t)\in {\mathbb{R}}^n\times (0,\infty )\}$, with uniformly complex elliptic, $L^{\infty }$, $t$-independent coefficients. By a standard pull-back mechanism, these results extend corresponding results of Kenig, Lin and Shen for the Laplacian in a Lipschitz domain, which have application to the theory of homogenization.

Some typos appearing in version 1 have been corrected in this version