In this paper we prove the generation of positive and analytic semigroups in $L^p(\R^N), 1<p<\infty$, for elliptic partial differential operators of the form $Au=\nabla(a\nabla u)+F\cdot \nabla u -Vu,$ where $a\in C^1(\R^N,\R^{N^2})$ is assumed to be bounded and uniformly elliptic, $F\in C^1(\R^N,\R^N)$ and $V:\R^N\to [0,\infty)$ satisfying $U\le V\le c_1U, |F|\le \kappa U^{1/2}$ where $U$ is a uniformly positive $C^1$-function such that $|\nabla U|\le \gamma U^{3/2}+C_\gamma$ for a sufficiently small $\gamma >0$ and $\theta U+div F\ge 0$ for some $\theta <p$. Analogous results are also established in the spaces $L^1(\R^N)$ and $C_0(\R^N)$. The proofs are based essentially on an interpolation inequality between $U$ and $U^{1/2}$. As an application we show that the generalized Ornstein-Uhlenbeck operator $Lu=\Delta u-\nabla \Phi \cdot \nabla u+G\cdot \nabla u$ with domain $W^{2,p}(\R^N,\mu)$ generates an analytic semigroup on the weighted space $L^p(\R^N,\mu), 1<p<\infty$, where $\mu(dx)=e^{-\Phi(x)}dx$.

L_p regularity for elliptic operators with unbounded coefficients

RHANDI, Abdelaziz;
2005-01-01

Abstract

In this paper we prove the generation of positive and analytic semigroups in $L^p(\R^N), 10$ and $\theta U+div F\ge 0$ for some $\theta
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/1658823
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