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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: April 19, 2005
Erratum: Proc. Amer. Math. Soc. 134 (2006), 2479-2480.
MathSciNet review: 2146208
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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
Email: lunardi@unipr.it

G. Metafune
Affiliation: Dipartimento di Matematica “Ennio De Giorgi”, Università di Lecce, C.P.193, 73100, Lecce, Italy
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

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08068-8
PII: S 0002-9939(05)08068-8
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