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

Robert Denk; Matthias Georg Hieber; Jan Prüss

doi:10.1007/s00209-007-0120-9

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 journal › Article › peer-review

179 Citations (Scopus)

Abstract

In this paper we investigate vector-valued parabolic initial boundary value problems A(t,x,D), B_j(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... 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.

Original language	English
Pages (from-to)	193-224
Number of pages	32
Journal	Mathematische Zeitschrift
Volume	257
Issue number	1
DOIs	https://doi.org/10.1007/s00209-007-0120-9
Publication status	Published - 2007 Sept
Externally published	Yes

Keywords

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

ASJC Scopus subject areas

Mathematics(all)

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10.1007/s00209-007-0120-9

Cite this

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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.",

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AU - Denk, Robert

AU - Hieber, Matthias Georg

AU - Prüss, Jan

PY - 2007/9

Y1 - 2007/9

N2 - 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.

AB - 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.

KW - Optimal L -L -estimates

KW - Parabolic boundary value problems with general boundary conditions

KW - Vector-valued Sobolev spaces

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

Abstract

Keywords

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