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The Green Function for Elliptic Systems in the Upper-Half Space

Martin Dindoš, Dorina Mitrea, Irina Mitrea, Marius Mitrea

Abstract

Let $L$ be a second-order, homogeneous, constant (complex) coefficient elliptic system in ${\mathbb{R}}^n$. The goal of this article is provide a qualitative and quantitative study of the nature of the Green function associated with the system $L$ in the upper-half space. Starting with a definition of the Green function which brings forth the minimal features which identify this object uniquely, we establish optimal nontangential maximal function estimates and regularity results up to the boundary for the said Green function. The main tools employed in the proof include the Agmon-Douglis-Nirenberg construction of a Poisson kernel for the system $L$, the Agmon-Douglis-Nirenberg a priori regularity estimates near the boundary, and the brand of Divergence Theorem from the book Geometric Harmonic Analysis Vol. I by the last three authors of this paper in which the boundary trace of the corresponding vector field is taken in nontangential pointwise sense.

The Green Function for Elliptic Systems in the Upper-Half Space

Abstract

Let be a second-order, homogeneous, constant (complex) coefficient elliptic system in . The goal of this article is provide a qualitative and quantitative study of the nature of the Green function associated with the system in the upper-half space. Starting with a definition of the Green function which brings forth the minimal features which identify this object uniquely, we establish optimal nontangential maximal function estimates and regularity results up to the boundary for the said Green function. The main tools employed in the proof include the Agmon-Douglis-Nirenberg construction of a Poisson kernel for the system , the Agmon-Douglis-Nirenberg a priori regularity estimates near the boundary, and the brand of Divergence Theorem from the book Geometric Harmonic Analysis Vol. I by the last three authors of this paper in which the boundary trace of the corresponding vector field is taken in nontangential pointwise sense.
Paper Structure (3 sections, 18 theorems, 276 equations)

This paper contains 3 sections, 18 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of the Main Results

Key Result

Theorem 1.2

Fix $n,M\in{\mathbb{N}}$ with $n\geq 2$. Assume $L$ is an $M\times M$ system with constant complex coefficients as in L-def-L-ell.X. Then there exists a unique Green function $G^L(\cdot,\cdot)$ for $L$ in $\mathbb{R}^n_{+}$, in the sense of Definition ta.av-GGG. Moreover, this Green function also sa

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • ...and 21 more