We show that the realization $A_p$ of the elliptic operator $\mathcal{A}u=div(Q\nabla u)+ F\cdot \nabla u+Vu$ in $L^p(\R^N,\R^N), 1\le p<\infty$, generates a strongly continuous semigroup, and we determine its domain $D(A_p)=\{u\in W^{2,p}(\R^N,\R^N): F\cdot \nabla u+Vu\in L^p(\R^N,\R^N)\} if $1<p<\infty$. The diffusion coefficients $Q =(qij)$ are uniformly elliptic and bounded together with their first-order derivatives, the drift coefficients F can grow as $|x|log|x|$, and V can grow logarithmically. Our approach relies on the Monniaux-Prüss theorem on the sum of non commuting operators. We also prove Lp-Lq estimates and, under somewhat stronger assumptions, we establish pointwise gradient estimates and smoothing of the semigroup in the spaces $W^{\alpha ,p}(\R^N,\R^N), \alpha \in [0,1], 1<p<\infty$.

Global properties of generalized Ornstein-Uhlenbeck operators on Lp(RN;RN) with more than linearly growing coefficients

RHANDI, Abdelaziz;
2009-01-01

Abstract

We show that the realization $A_p$ of the elliptic operator $\mathcal{A}u=div(Q\nabla u)+ F\cdot \nabla u+Vu$ in $L^p(\R^N,\R^N), 1\le p<\infty$, generates a strongly continuous semigroup, and we determine its domain $D(A_p)=\{u\in W^{2,p}(\R^N,\R^N): F\cdot \nabla u+Vu\in L^p(\R^N,\R^N)\} if $1
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/2292181
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