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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Pseudo-differential operators
and maximal regularity results
for non-autonomous parabolic equations


Authors: Matthias Hieber and Sylvie Monniaux
Journal: Proc. Amer. Math. Soc. 128 (2000), 1047-1053
MSC (1991): Primary 35K22, 35S05, 47D06
Published electronically: July 28, 1999
MathSciNet review: 1641630
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Abstract: In this paper, we show that a pseudo-differential operator associated to a symbol $a\in L^{\infty}(\mathbb{R}\times\mathbb{R},\mathcal{L}(H)) $ ($H$ being a Hilbert space) which admits a holomorphic extension to a suitable sector of $\mathbb{C}$ acts as a bounded operator on $L^{2}(\mathbb{R},H)$. By showing that maximal $L^{p}$-regularity for the non-autonomous parabolic equation $u'(t) + A(t)u(t) = f(t), u(0)=0$ is independent of $p\in (1,\infty)$, we obtain as a consequence a maximal $L^{p}([0,T],H)$-regularity result for solutions of the above equation.


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Additional Information

Matthias Hieber
Affiliation: Mathematisches Institut I, Englerstr. 2, Universität Karlsruhe, D-76128 Karlsruhe, Germany
Email: matthias.hieber@math.uni-karlsruhe.de

Sylvie Monniaux
Affiliation: Abteilung Mathematik V, Universität Ulm, D-89069 Ulm, Germany
Address at time of publication: Laboratoire de Mathématiques Fondamentales et Appliquées, Centre de Saint-Jérôme, Case Cour A, Avenue Escadrille Normandie-Niemen, 13397 Marseille Cédex 20, France
Email: monniaux@mathematik.uni-ulm.de, sylvie.monniaux@math.u-3mrs.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05145-X
PII: S 0002-9939(99)05145-X
Received by editor(s): May 18, 1998
Published electronically: July 28, 1999
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society