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

• 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.
Mathematics Subject Classification: Primary: 47D07; Secondary: 35B50, 35J25, 35J70.

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