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

### Peter Weidemaier

**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.

*Copyright 2002 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**08** (2002), pp. 47-51
- Publisher Identifier: S 1079-6762(02)00104-X
- 2000
*Mathematics Subject Classification*. Primary 35K20, 46E35; Secondary 26D99
*Key words and phrases*. Maximal regularity, inhomogeneous boundary conditions, trace theory, mixed norm, Lizorkin-Triebel spaces
- Received by editors October 16, 2002
- Posted on December 19, 2002
- Communicated by Michael E. Taylor
- Comments (When Available)

**Peter Weidemaier**

Fraunhofer-Institut Kurzzeitdynamik, Eckerstr. 4, D-79104 Freiburg, Germany

*E-mail address:* `weide@emi.fhg.de`

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