On Poisson operators and Dirichlet-Neumann maps in H^s for divergence form elliptic operators with Lipschitz coefficients

TL;DR

This paper characterizes the domain of Poisson operators and Dirichlet-Neumann maps in Sobolev spaces for divergence form elliptic operators with Lipschitz coefficients, providing a factorization formula for the elliptic operator.

Contribution

It offers a new characterization of these operators' domains and a factorization formula, extending understanding in elliptic PDEs with Lipschitz coefficients.

Findings

Characterization of Poisson operators in H^s spaces for s in [0,1]

Description of Dirichlet-Neumann maps in the same Sobolev spaces

A factorization formula for the elliptic operator

Abstract

We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. Under the Lipschitz condition on the coefficients we characterize the domain of the Poisson operators and the Dirichlet-Neumann maps in the Sobolev space $H^s(\R^d)$ for each $s\in [0,1]$. Moreover, we also show a factorization formula for the elliptic operator in terms of the Poisson operator.