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Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm

Author(s): Peter Weidemaier
Journal: Electron. Res. Announc. Amer. Math. Soc. 8 (2002), 47-51.
MSC (2000): Primary 35K20, 46E35; Secondary 26D99
Posted: December 19, 2002
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Abstract: We determine the exact regularity of the trace of a function $ u \in L_{q}\,(0,T;\, W_{p}^{2}(\Omega)) $ $ \cap \, W^{1}_{q}\,(0,T;\, {L_{p}\,(\Omega))} $ and of the trace of its spatial gradient on $\partial \Omega \times (\,0,T\,) $ in the regime $ p \le q $. While for $ p=q $ both the spatial and temporal regularity of the traces can be completely characterized by fractional order Sobolev-Slobodetskii spaces, for $ p \neq q $ the Lizorkin-Triebel spaces turn out to be necessary for characterizing the sharp temporal regularity.

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Additional Information:

Peter Weidemaier
Affiliation: Fraunhofer-Institut Kurzzeitdynamik, Eckerstr. 4, D-79104 Freiburg, Germany
Email: weide@emi.fhg.de

DOI: 10.1090/S1079-6762-02-00104-X
PII: S 1079-6762(02)00104-X
Keywords: Maximal regularity, inhomogeneous boundary conditions, trace theory, mixed norm, Lizorkin-Triebel spaces
Received by editor(s): October 16, 2002
Posted: December 19, 2002
Communicated by: Michael E. Taylor
Copyright of article: Copyright 2002, American Mathematical Society

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Kozowski,Z. and Pawlow,I., Unique global solvability of the Fried-Gurtin model for phase transitions in solids, Topol. Methods Nonlinear Analysis 24 (2004), 209-237. (English)

Peter Weidemaier, Vector-valued Lizorkin-Triebel spaces and sharp trace theory for functions in Sobolev spaces with mixed L_p norm for parabolic problems, Sb. Math. (6) 196 (2005), 777-790. (English) MR 2164549

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