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

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 ) with bounded geometry. We employ an...

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Bibliographic Details
Published inJournal of evolution equations Vol. 17; no. 1; pp. 51 - 100
Main Author Amann, Herbert
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.03.2017
Springer Nature B.V
Subjects
Analysis
Mathematics
Mathematics and Statistics
Parabolic differential equations
35K52
35K51
58J99
Parabolic initial value problems
Hölder spaces
non-compact manifolds
Sobolev–Slobodeckii spaces
maximal regularity
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Summary: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 ) 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 = ( R m , | d x | 2 ) , we recover classical results for Petrowskii-parabolic Cauchy problems.
ISSN:1424-3199
1424-3202
DOI:10.1007/s00028-016-0347-1