### $L^p$-regularity for elliptic operators with unbounded coefficients

#### 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 $\mu(dx)=e^{-\Phi(x)}dx$.

#### Article information

Source
Adv. Differential Equations Volume 10, Number 10 (2005), 1131-1164.

Dates
First available in Project Euclid: 18 December 2012

Metafune, G.; Prüss, J.; Schnaubelt, R.; Rhandi, A. $L^p$-regularity for elliptic operators with unbounded coefficients. Adv. Differential Equations 10 (2005), no. 10, 1131--1164. https://projecteuclid.org/euclid.ade/1355867807.