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Cauchy problems for parabolic equations in Sobolev–Slobodeckii and Hölder spaces on uniformly regular Riemannian manifolds

Cauchy problems for parabolic equations in Sobolev–Slobodeckii and Hölder spaces on uniformly... In this paper, we establish optimal solvability results—maximal regularity theorems—for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless complete Riemannian manifolds $$({M, g})$$ ( M , g ) with bounded geometry. We employ an anisotropic extension of the Fourier multiplier theorem for arbitrary Besov spaces introduced in Amann (Math Nachr 186:5–56, 1997). This allows for a unified treatment of Sobolev–Slobodeckii and little Hölder spaces. In the flat case $${(M, g=(\mathbb{R}^{m},|dx|^{2})}$$ ( M , g = ( R m , | d x | 2 ) , we recover classical results for Petrowskii-parabolic Cauchy problems. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Cauchy problems for parabolic equations in Sobolev–Slobodeckii and Hölder spaces on uniformly regular Riemannian manifolds

Journal of Evolution Equations , Volume 17 (1) – Sep 27, 2016

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References (54)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-016-0347-1
Publisher site
See Article on Publisher Site

Abstract

In this paper, we establish optimal solvability results—maximal regularity theorems—for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless complete Riemannian manifolds $$({M, g})$$ ( M , g ) with bounded geometry. We employ an anisotropic extension of the Fourier multiplier theorem for arbitrary Besov spaces introduced in Amann (Math Nachr 186:5–56, 1997). This allows for a unified treatment of Sobolev–Slobodeckii and little Hölder spaces. In the flat case $${(M, g=(\mathbb{R}^{m},|dx|^{2})}$$ ( M , g = ( R m , | d x | 2 ) , we recover classical results for Petrowskii-parabolic Cauchy problems.

Journal

Journal of Evolution EquationsSpringer Journals

Published: Sep 27, 2016

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