On domain of Poisson operators and factorization for divergence form elliptic operators

TL;DR

This paper studies divergence form elliptic operators with coefficients independent of one variable, providing a factorization into first order operators, a solution formula for boundary problems, and establishing $L^2$ solvability.

Contribution

It introduces a novel factorization approach for elliptic operators with specific coefficient structures, enabling new solution formulas and solvability results.

Findings

Factorization into first order operators related to Poisson and Dirichlet-Neumann maps

Explicit solution formula for inhomogeneous boundary value problems

Proved $L^2$ solvability for a new class of elliptic operators

Abstract

We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators related with Poisson operators and Dirichlet-Neumann maps. Consequently, we obtain a solution formula for the inhomogeneous elliptic boundary value problem in the half space, which is useful to show the existence of solutions in a wider class of inhomogeneous data. We also establish $L^2$ solvability of boundary value problems for a new class of the elliptic operators.