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

TL;DR

This paper proves new $L^2$ estimates for commutators of the Dirichlet-to-Neumann map related to complex elliptic operators in the upper half-space, extending prior results and aiding homogenization theory.

Contribution

It establishes $L^2$ commutator estimates for the Dirichlet-to-Neumann map with complex coefficients, generalizing previous Laplacian results to more complex elliptic operators.

Findings

Established $L^2$ bounds for commutators of the Dirichlet-to-Neumann map.

Extended previous Laplacian results to complex elliptic operators.

Applications to homogenization theory and elliptic PDE analysis.

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.