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

Simona Fornaro; Luca Lorenzi

doi:10.3934/dcds.2007.18.747

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2007, Volume 18, Issue 4: 747-772. Doi: 10.3934/dcds.2007.18.747

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

Simona Fornaro^1, and
Luca Lorenzi^2,

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

Received: October 31, 2006

Revised: December 31, 2006

Published: September 2007

Abstract Related Papers Cited by

Abstract

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.

Keywords:

degenerate and nondegenerate elliptic operators,
unbounded coefficients,
exterior domains,
generation results,
analytic semigroups.

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

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