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The solvability and subellipticity of systems of pseudodifferential operators

Dencker, Nils LU (2009) Siena workshop in honor of Ferruccio Colombini on the occasion of his 60th birthday, 2007 In Progress in Nonlinear Differential Equations and Their Applications 78. p.73-94
Abstract

The paper studies the local solvability and subellipticity for square systems of principal type. These are the systems for which the principal symbol vanishes of first order on its kernel. For systems of principal type having constant characteristics, local solvability is equivalent to condition (Ψ) on the eigenvalues. This is a condition on the sign changes of the imaginary part along the oriented bicharacteristics of the real part of the eigenvalue. In the generic case when the principal symbol does not have constant characteristics, condition (Ψ) is not sufficient and in general not well defined. Instead we study systems which are quasi-symmetrizable, these systems have natural invariance properties and are of principal type. We... (More)

The paper studies the local solvability and subellipticity for square systems of principal type. These are the systems for which the principal symbol vanishes of first order on its kernel. For systems of principal type having constant characteristics, local solvability is equivalent to condition (Ψ) on the eigenvalues. This is a condition on the sign changes of the imaginary part along the oriented bicharacteristics of the real part of the eigenvalue. In the generic case when the principal symbol does not have constant characteristics, condition (Ψ) is not sufficient and in general not well defined. Instead we study systems which are quasi-symmetrizable, these systems have natural invariance properties and are of principal type. We prove that quasi-symmetrizable systems are locally solvable. We also study the subellipticity of quasi-symmetrizable systems in the case when principal symbol vanishes of finite order along the bicharacteristics. In order to prove subellipticity, we assume that the principal symbol has the approximation property, which implies that there are no transversal bicharacteristics.

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Please use this url to cite or link to this publication:
author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
Principal type, Pseudodifferential, Solvability, Subelliptic, System
host publication
Advances in Phase Space Analysis of Partial Differential Equations - In Honor of Ferruccio Colombini's 60th Birthday
series title
Progress in Nonlinear Differential Equations and Their Applications
editor
Del Santo, Daniele ; Murthy, M.K. Venkatesha and Bove, Antonio
volume
78
pages
22 pages
publisher
Springer
conference name
Siena workshop in honor of Ferruccio Colombini on the occasion of his 60th birthday, 2007
conference location
Siena, Italy
conference dates
2007-10-10 - 2007-10-13
external identifiers
  • scopus:84877910155
ISSN
2374-0280
1421-1750
ISBN
9780817648602
DOI
10.1007/978-0-8176-4861-9_5
language
English
LU publication?
yes
id
fff2fb03-1275-4f64-baae-b9c3ba6f3139
date added to LUP
2019-06-24 10:49:47
date last changed
2021-06-16 06:09:28
@inproceedings{fff2fb03-1275-4f64-baae-b9c3ba6f3139,
  abstract     = {<p>The paper studies the local solvability and subellipticity for square systems of principal type. These are the systems for which the principal symbol vanishes of first order on its kernel. For systems of principal type having constant characteristics, local solvability is equivalent to condition (Ψ) on the eigenvalues. This is a condition on the sign changes of the imaginary part along the oriented bicharacteristics of the real part of the eigenvalue. In the generic case when the principal symbol does not have constant characteristics, condition (Ψ) is not sufficient and in general not well defined. Instead we study systems which are quasi-symmetrizable, these systems have natural invariance properties and are of principal type. We prove that quasi-symmetrizable systems are locally solvable. We also study the subellipticity of quasi-symmetrizable systems in the case when principal symbol vanishes of finite order along the bicharacteristics. In order to prove subellipticity, we assume that the principal symbol has the approximation property, which implies that there are no transversal bicharacteristics.</p>},
  author       = {Dencker, Nils},
  booktitle    = {Advances in Phase Space Analysis of Partial Differential Equations - In Honor of Ferruccio Colombini's 60th Birthday},
  editor       = {Del Santo, Daniele and Murthy, M.K. Venkatesha and Bove, Antonio},
  isbn         = {9780817648602},
  issn         = {2374-0280},
  language     = {eng},
  month        = {08},
  pages        = {73--94},
  publisher    = {Springer},
  series       = {Progress in Nonlinear Differential Equations and Their Applications},
  title        = {The solvability and subellipticity of systems of pseudodifferential operators},
  url          = {http://dx.doi.org/10.1007/978-0-8176-4861-9_5},
  doi          = {10.1007/978-0-8176-4861-9_5},
  volume       = {78},
  year         = {2009},
}