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Discrete Fourier multipliers and cylindrical boundary-value problems

Published online by Cambridge University Press:  03 December 2013

Robert Denk  and
Tobias Nau
Robert Denk
Affiliation:
Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany ([email protected]; [email protected])
Tobias Nau
Affiliation:
Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany ([email protected]; [email protected])
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Abstract

We consider operator-valued boundary-value problems in (0, 2π)n with periodic or, more generally, ν-periodic boundary conditions. Using the concept of discrete vector-valued Fourier multipliers, we give equivalent conditions for the unique solvability of the boundary-value problem. As an application, we study vector-valued parabolic initial boundary-value problems in cylindrical domains (0, 2π)n × V with ν-periodic boundary conditions in the cylindrical directions. We show that, under suitable assumptions on the coefficients, we obtain maximal Lq-regularity for such problems. For symmetric operators such as the Laplacian, related results for mixed Dirichlet-Neumann boundary conditions on (0, 2π)n × V are deduced.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 143 , Issue 6 , December 2013 , pp. 1163 - 1183
Copyright
Copyright © Royal Society of Edinburgh 2013 

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