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Proceedings of the American Mathematical Society

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Dirichlet boundary conditions for elliptic operators with unbounded drift


Authors: A. Lunardi, G. Metafune and D. Pallara
Journal: Proc. Amer. Math. Soc. 133 (2005), 2625-2635
MSC (2000): Primary 35J70; Secondary 47D07
DOI: https://doi.org/10.1090/S0002-9939-05-08068-8
Published electronically: April 19, 2005
Erratum: Proc. Amer. Math. Soc. 134 (2006), 2479-2480.
MathSciNet review: 2146208
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the realisation $A$ of the operator $\mathcal {A} = \Delta - \langle D\Phi , D\cdot \rangle$ in $L^2(\Omega , \mu )$ with Dirichlet boundary condition, where $\Omega$ is a possibly unbounded open set in $\mathbb {R}^N$, $\Phi$ is a semi-convex function and the measure $d\mu (x) = \exp (-\Phi (x)) dx$ lets $\mathcal {A}$ be formally self-adjoint. The main result is that $A:D(A)= \{u\in H^2(\Omega , \mu ): \langle D\Phi , Du \rangle \in L^2(\Omega , \mu ), u=0$ at $\partial \Omega \}$ is a dissipative self-adjoint operator in $L^2(\Omega , \mu )$.


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Additional Information

A. Lunardi
Affiliation: Dipartimento di Matematica, Università di Parma, Parco Area delle Scienze 53, 43100 Parma, Italy
MR Author ID: 116935
Email: lunardi@unipr.it

G. Metafune
Affiliation: Dipartimento di Matematica “Ennio De Giorgi”, Università di Lecce, C.P.193, 73100, Lecce, Italy
MR Author ID: 123880
Email: giorgio.metafune@unile.it

D. Pallara
Affiliation: Dipartimento di Matematica “Ennio De Giorgi”, Università di Lecce, C.P.193, 73100, Lecce, Italy
Email: diego.pallara@unile.it

Keywords: Elliptic operators, boundary value problems, unbounded coefficients
Received by editor(s): April 19, 2004
Published electronically: April 19, 2005
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2005 American Mathematical Society