# American Institute of Mathematical Sciences

November  2007, 18(4): 747-772. doi: 10.3934/dcds.2007.18.747

## Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces

 1 Dipartimento di Matematica “Ennio De Giorgi”, Università degli Studi di Lecce, C.P. 193, I-73100 Lecce, Italy 2 Dipartimento di Matematica, Universitá degli Studi di Parma, Viale G. Usberti 85/A, 43100 Parma, Italy

Received  November 2006 Revised  January 2007 Published  May 2007

Let $a$ and $b$ be unbounded functions in $\mathbb R^N$ with $a$ sufficiently smooth. In this paper we prove that, under suitable growth assumptions on $a$ and $b$, the operator $Au=a\Delta u+b\cdot\nabla u$ admits realizations generating analytic semigroups in $L^p( R^N)$ for any $p\in [1,+\infty]$ and in $C_b( R^N)$. We also explicitly characterize the domain of the infinitesimal generator of such semigroups. Similar results are stated and proved when $R^N$ is replaced with a smooth exterior domain under general boundary conditions.
Citation: Simona Fornaro, Luca Lorenzi. Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 747-772. doi: 10.3934/dcds.2007.18.747
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