Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems
type: Item
instance of: scholarly article
title: Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems
zbMATH Open document ID: 1157.35028
author: Pascal Auscher
author: Andreas Axelsson
author: Steve Hofmann
published in: Journal of Functional Analysis
publication date: 2008-07-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/0705.0250
review text: The Neumann, Dirichlet and regularity problems are studied in \(L^2(\mathbb R^n)\) for divergence form second order elliptic equations \(\text{div}_{t,x}A(x)\nabla_{t,x}U(t,x)=0\) on the half-space \(\mathbb R_+^{n+1}=\{ (t,x)\in \mathbb R\times \mathbb R^n; t>0\} \). Let \(A_0(x)=((a_0)_{ij}(x))\) (\(i,j=0,\dots ,n\)) be a \(t\)-independent complex, accretive coefficient matrix function with \((a_0)_{ij}\in L^\infty (\mathbb R^n)\). Furthermore assume that \(A_0(x)\) is either real symmetric of block form or constant. Then there exists \(\epsilon >0\) depending only on \(\| A\| _\infty \), the accretivity constant and the dimension \(n\), such that if \(A\in L_\infty (\mathbb R^n; L(C^{n+1}))\) is \(t\)-independent and satisfies \(\| A-A_0 \| _\infty <\epsilon \), then Neumann, Dirichlet and regularity problems are well posed in \(L_2\). The problems are studied by a new boundary operator method. It is also proved that certain transmission problems are well posed.
reviewed by: Dagmar Medková
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 31B10
zbMATH DE Number: 5306470
zbMATH Keywords: Neumann problem
zbMATH Keywords: dirichlet problem
zbMATH Keywords: elliptic equation
zbMATH Keywords: non-symetric coefficients
zbMATH Keywords: dirac operator
zbMATH Keywords: functional calculus
zbMATH Keywords: quadratic estimates
zbMATH Keywords: perturbation theory
zbMATH Keywords: Carleson measure
MaRDI profile type: Publication
OpenAlex ID: W2963886478
arXiv ID: 0705.0250
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DOI: 10.1016/J.JFA.2008.02.007
description: scientific article