Optimal L p -L q -estimates for parabolic boundary value problems with inhomogeneous data

Robert Denk, Matthias Georg Hieber*, Jan Prüss

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

201 Citations (Scopus)


In this paper we investigate vector-valued parabolic initial boundary value problems A(t,x,D), Bj(t,x,D) subject to general boundary conditions in domains G in ℝn with compact C 2m -boundary. The top-order coefficients of A are assumed to be continuous. We characterize optimal L p -L q -regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on A and the Lopatinskii-Shapiro condition on A, B1... Bm) are necessary for these L p -L q -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.

Original languageEnglish
Pages (from-to)193-224
Number of pages32
JournalMathematische Zeitschrift
Issue number1
Publication statusPublished - 2007 Sept
Externally publishedYes


  • Optimal L -L -estimates
  • Parabolic boundary value problems with general boundary conditions
  • Vector-valued Sobolev spaces

ASJC Scopus subject areas

  • General Mathematics


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