Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm
Author:
Peter Weidemaier
Journal:
Electron. Res. Announc. Amer. Math. Soc. 8 (2002), 47-51
MSC (2000):
Primary 35K20, 46E35; Secondary 26D99
DOI:
https://doi.org/10.1090/S1079-6762-02-00104-X
Published electronically:
December 19, 2002
MathSciNet review:
1945779
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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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BeIN78 Besov O. V., Il’in V. P., Nikol’skii S. M., Integral representations of functions and imbedding theorems, vol. 1, Wiley, New York, 1978.
CaV86 Cannarsa P., Vespri V., On maximal $L^p$ regularity for the abstract Cauchy problem, Boll. Unione Mat. Ital. (6) 9–B (1986), 165–175.
ClP92 Clément Ph., Prüss J., Global existence for a semilinear parabolic Volterra equation, Math. Z. 209 (1992), 17–26.
CoD00 Coulhon Th., Duong X. T., Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss, Advances Differential Equations, 5 (2000), 343–368.
Gri66 Grisvard P., Commutativité de deux foncteurs d’interpolation et applications, II, J. Math. Pures Appl. 45 (1966), 207–290.
HiP97 Hieber M., Prüss J., Heat kernels and maximal $L^p-L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations 22 (1997), 1647–1669.
IlS62 Il’in V. P., Solonnikov V. A., On some properties of differentiable functions of several variables, Transl. AMS 81 (1969), 67–90 (Trudy Mat. Inst. Steklov 66 (1962), 205–226).
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Soh01 Sohr H., The Navier-Stokes equations, Birkhäuser, Basel, 2001.
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Wei94 Weidemaier P., On the trace theory for functions in Sobolev spaces with mixed $L_p$-norm, Czechoslovak Math. J. 44 (1994), 7–20.
Wei95 Weidemaier P., On the sharp initial trace of functions with derivatives in $L_q(0, T; L_p(\omega ))$, Boll. Unione Mat. Ital. 9-B (1995), 321–338.
Wei98 Weidemaier P., Existence results in $L_p-L_q$-spaces for second order parabolic equations with inhomogeneous boundary conditions, In: Amann H. et al., Progress in Partial Differential Equations, Proc. Pont-à-Mousson 1997, Pitman Research Notes in Mathematics 384, Longman, Harlow, UK, 1998, pp. 189–200.
Hab Weidemaier P., Maximal regularity results in Sobolev spaces with mixed $L_p$-norm for linear parabolic equations of second order with inhomogeneous boundary conditions, 1998 (unpublished manuscript).
WeS99 Weidemaier P., Sinnamon G., Perturbed weighted Hardy inequalities, Journal Math. Anal. Appl. 234 (1999), 287–292.
PWe12Weidemaier P., Vector-valued Lizorkin-Triebel spaces and sharp trace theory for functions in Sobolev spaces with mixed $L_p$-norm for parabolic problems, submitted.
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Additional Information
Peter Weidemaier
Affiliation:
Fraunhofer-Institut Kurzzeitdynamik, Eckerstr. 4, D-79104 Freiburg, Germany
Email:
weide@emi.fhg.de
Keywords:
Maximal regularity,
inhomogeneous boundary conditions,
trace theory,
mixed norm,
Lizorkin-Triebel spaces
Received by editor(s):
October 16, 2002
Published electronically:
December 19, 2002
Communicated by:
Michael E. Taylor
Article copyright:
© Copyright 2002
American Mathematical Society
