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$L^p-L^q$ ESTIMATES FOR PARABOLIC SYSTEMS IN NON-DIVERGENCE FORM WITH VMO COEFFICIENTS

Published online by Cambridge University Press:  04 January 2007

ROBERT HALLER-DINTELMANN
Affiliation:
Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstraße 7, D-64289 Darmstadt, Germanyhaller@mathematik.tu-darmstadt.de
HORST HECK
Affiliation:
Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstraße 7, D-64289 Darmstadt, Germanyheck@mathematik.tu-darmstadt.de
MATTHIAS HIEBER
Affiliation:
Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstraße 7, D-64289 Darmstadt, Germanyhieber@mathematik.tu-darmstadt.de
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Abstract

Consider a parabolic $N\times N$-system of order $m$ on $\mathbb{R}^n$ with top-order coefficients $a_\alpha \in \mathrm{VMO} \cap L^\infty$. Let $1<p,q < \infty$ and let $\omega$ be a Muckenhoupt weight. It is proved that systems of this kind possess a unique solution $u$ satisfying

$$\|u'\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)} + \|\mathcal{A} u\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)} \le C \|f\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)},$$

where $\mathcal{A} u = \sum_{|\alpha| \le m}a_\alpha D^\alpha u$ and $J=[0,\infty)$. In particular, choosing $\omega =1$, the realization of $\mathcal{A}$ in $L^p({\mathbb{R}}^n)^N$ has maximal $L^p-L^q$ regularity.

Type
Notes and Papers
Copyright
The London Mathematical Society 2006

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Footnotes

Supported by the Deutsche Forschungsgemeinschaft (DFG) by the project ‘Regularity properties of elliptic and parabolic equations’.