Mathematische Zeitschrift

, Volume 257, Issue 1, pp 193–224

Optimal Lp-Lq-estimates for parabolic boundary value problems with inhomogeneous data

Original Paper

DOI: 10.1007/s00209-007-0120-9

Cite this article as:
Denk, R., Hieber, M. & Prüss, J. Math. Z. (2007) 257: 193. doi:10.1007/s00209-007-0120-9


In this paper we investigate vector-valued parabolic initial boundary value problems \({(\mathcal A(t,x,D)}\) , \({\mathcal B_j(t,x,D))}\) subject to general boundary conditions in domains G in \({\mathbb R^n}\) with compact C2m-boundary. The top-order coefficients of \({\mathcal A}\) are assumed to be continuous. We characterize optimal Lp-Lq-regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on \({\mathcal A}\) and the Lopatinskii–Shapiro condition on \({(\mathcal A, \mathcal B_1,\dots, \mathcal B_m)}\) are necessary for these Lp-Lq-estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.


Parabolic boundary value problems with general boundary conditionsOptimal Lp-Lq-estimatesVector-valued Sobolev spaces

Mathematics Subject Classification (1991)


Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Fachbereich Mathematik und StatistikUniversität KonstanzKonstanzGermany
  2. 2.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  3. 3.Fachbereich Mathematik und Informatik, Institut für AnalysisMartin-Luther-Universität Halle-WittenbergHalleGermany