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

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Abstract

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 C 2m-boundary. The top-order coefficients of \({\mathcal A}\) 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 \({\mathcal A}\) and the Lopatinskii–Shapiro condition on \({(\mathcal A, \mathcal B_1,\dots, \mathcal B_m)}\) 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.

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Authors and Affiliations

  1. Fachbereich Mathematik und Statistik, Universität Konstanz, 78457, Konstanz, Germany

    Robert Denk

  2. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Matthias Hieber

  3. Fachbereich Mathematik und Informatik, Institut für Analysis, Martin-Luther-Universität Halle-Wittenberg, Theodor-Lieser-Str. 5, 06120, Halle, Germany

    Jan Prüss

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  1. Robert Denk
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  2. Matthias Hieber
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Correspondence to Matthias Hieber.

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Denk, R., Hieber, M. & Prüss, J. Optimal L p-L q-estimates for parabolic boundary value problems with inhomogeneous data. Math. Z. 257, 193–224 (2007). https://doi.org/10.1007/s00209-007-0120-9

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  • DOI: https://doi.org/10.1007/s00209-007-0120-9

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