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