R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type

Robert Denk, Matthias Georg Hieber, Jan Prüss

Research output: Contribution to journalArticle

418 Citations (Scopus)

Abstract

The property of maximal Lp-regularity for parabolic evolution equations is investigated via the concept of R-sectorial operators and operator-valued Fourier multipliers. As application, we consider the L q-realization of an elliptic boundary value problem of order 2m with operator-valued coefficients subject to general boundary conditions. We show that there is maximal Lp-Lq-regularity for the solution of the associated Cauchy problem provided the top order coefficients are bounded and uniformly continuous.

Original languageEnglish
JournalMemoirs of the American Mathematical Society
Issue number788
Publication statusPublished - 2003 Nov
Externally publishedYes

Fingerprint

R-boundedness
Fourier multipliers
Boundary value problems
Mathematical operators
Operator-valued Fourier multipliers
Regularity
Boundary conditions
Sectorial Operator
General Boundary Conditions
Uniformly continuous
Elliptic Boundary Value Problems
Coefficient
Parabolic Equation
Evolution Equation
Cauchy Problem
Operator

Keywords

  • Differential operators with operator-valued coefficients
  • L-theory for boundary value problems of general type
  • Lopatinskii-Shapiro condition
  • Maximal L-regularity
  • Operator-valued Fourier multipliers
  • R-boundedness

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type. / Denk, Robert; Hieber, Matthias Georg; Prüss, Jan.

In: Memoirs of the American Mathematical Society, No. 788, 11.2003.

Research output: Contribution to journalArticle

@article{1a3c077b375640ab886b4ddd432adc06,
title = "R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type",
abstract = "The property of maximal Lp-regularity for parabolic evolution equations is investigated via the concept of R-sectorial operators and operator-valued Fourier multipliers. As application, we consider the L q-realization of an elliptic boundary value problem of order 2m with operator-valued coefficients subject to general boundary conditions. We show that there is maximal Lp-Lq-regularity for the solution of the associated Cauchy problem provided the top order coefficients are bounded and uniformly continuous.",
keywords = "Differential operators with operator-valued coefficients, L-theory for boundary value problems of general type, Lopatinskii-Shapiro condition, Maximal L-regularity, Operator-valued Fourier multipliers, R-boundedness",
author = "Robert Denk and Hieber, {Matthias Georg} and Jan Pr{\"u}ss",
year = "2003",
month = "11",
language = "English",
journal = "Memoirs of the American Mathematical Society",
issn = "0065-9266",
publisher = "American Mathematical Society",
number = "788",

}

TY - JOUR

T1 - R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type

AU - Denk, Robert

AU - Hieber, Matthias Georg

AU - Prüss, Jan

PY - 2003/11

Y1 - 2003/11

N2 - The property of maximal Lp-regularity for parabolic evolution equations is investigated via the concept of R-sectorial operators and operator-valued Fourier multipliers. As application, we consider the L q-realization of an elliptic boundary value problem of order 2m with operator-valued coefficients subject to general boundary conditions. We show that there is maximal Lp-Lq-regularity for the solution of the associated Cauchy problem provided the top order coefficients are bounded and uniformly continuous.

AB - The property of maximal Lp-regularity for parabolic evolution equations is investigated via the concept of R-sectorial operators and operator-valued Fourier multipliers. As application, we consider the L q-realization of an elliptic boundary value problem of order 2m with operator-valued coefficients subject to general boundary conditions. We show that there is maximal Lp-Lq-regularity for the solution of the associated Cauchy problem provided the top order coefficients are bounded and uniformly continuous.

KW - Differential operators with operator-valued coefficients

KW - L-theory for boundary value problems of general type

KW - Lopatinskii-Shapiro condition

KW - Maximal L-regularity

KW - Operator-valued Fourier multipliers

KW - R-boundedness

UR - http://www.scopus.com/inward/record.url?scp=0242287337&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0242287337&partnerID=8YFLogxK

M3 - Article

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

SN - 0065-9266

IS - 788

ER -