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

Robert Denk, Matthias Georg Hieber, Jan Prüss

Research output: Contribution to journalArticle

119 Citations (Scopus)

Abstract

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
Volume257
Issue number1
DOIs
Publication statusPublished - 2007 Sep
Externally publishedYes

Fingerprint

Boundary Value Problem
Triebel-Lizorkin Space
General Boundary Conditions
Ellipticity
Parabolic Problems
Estimate
Initial-boundary-value Problem
Sobolev Spaces
Regularity
Trace
Necessary
Coefficient

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Optimal L p -L q -estimates for parabolic boundary value problems with inhomogeneous data. / Denk, Robert; Hieber, Matthias Georg; Prüss, Jan.

In: Mathematische Zeitschrift, Vol. 257, No. 1, 09.2007, p. 193-224.

Research output: Contribution to journalArticle

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