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2005 $L^p$-regularity for elliptic operators with unbounded coefficients
G. Metafune, J. Prüss, A. Rhandi, R. Schnaubelt
Adv. Differential Equations 10(10): 1131-1164 (2005).

Abstract

Under suitable conditions on the functions $a\in C^1({\mathbb R}^N,{\mathbb R}^{N^2})$, $F\in C^1({\mathbb R}^N,{\mathbb R}^N)$, and $V:{\mathbb R}^N\to [0,\infty)$, we show that the operator $Au=\nabla (a\nabla u) +F\cdot \nabla u-Vu$ with domain $W^{2,p}_V({\mathbb R}^N)= \{ u\in W^{2,p}({\mathbb R}^N):Vu\in L^p({\mathbb R}^N) \}$ generates a positive analytic semigroup on $L^p({\mathbb R}^N)$, $1 < p < \infty$. Analogous results are also established in the spaces $L^1({\mathbb R}^N)$ and $C_0({\mathbb R}^N)$. As an application we show that the generalized Ornstein--Uhlenbeck operator $A_{\Phi,G} u=\Delta u -\nabla \Phi \cdot \nabla u + G\cdot \nabla u$ with domain $W^{2,p}({\mathbb R}^N,\mu)$ generates an analytic semigroup on the weighted space $L^p({\mathbb R}^N,\mu)$, where $1 < p < \infty$ and $\mu(dx)=e^{-\Phi(x)}dx$.

Citation

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G. Metafune. J. Prüss. A. Rhandi. R. Schnaubelt. "$L^p$-regularity for elliptic operators with unbounded coefficients." Adv. Differential Equations 10 (10) 1131 - 1164, 2005.

Information

Published: 2005
First available in Project Euclid: 18 December 2012

zbMATH: 1156.35385
MathSciNet: MR2162364

Subjects:
Primary: 35J70
Secondary: 35K65, 47D06

Rights: Copyright © 2005 Khayyam Publishing, Inc.

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