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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: 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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  • 1. M. BERTOLDI, S. FORNARO: Gradient estimates in parabolic problems with unbounded coefficients, Studia Math. 165 (2004), 221-254. MR 2109509
  • 2. S. CERRAI: Second order PDE's in finite and infinite dimensions. A probabilistic approach, Lecture Notes in Mathematics 1762, Springer-Verlag, Berlin (2001). MR 1840644 (2002j:35327)
  • 3. G. DA PRATO, A. LUNARDI: Elliptic operators with unbounded drift coefficients and Neumann boundary condition, J. Diff. Eqns. 198 (2004), 35-52. MR 2037749 (2004k:35066)
  • 4. E. B. DAVIES: Heat Kernels and Spectral Theory, Cambridge U. P. (1989). MR 0990239 (90e:35123)
  • 5. L. GROSS: Logarithmic Sobolev Inequalities, Amer. J. Math. 97 (1975), 1061-1083. MR 0420249 (54:8263)
  • 6. L. GROSS: Logarithmic Sobolev Inequalities and Contractivity Properties of Semigroups, in: Dirichlet Forms (G. Dell'Antonio and U. Mosco Eds.), Lecture Notes in Mathematics 1563, Springer-Verlag, Berlin (1993), 54-88. MR 1292277 (95h:47061)
  • 7. G. METAFUNE, D. PALLARA, V. VESPRI: $L^p$-estimates for a class of elliptic operators with unbounded coefficients in $\mathbb{R} ^N$, Houston J. Math. 31 (2005), 605-620.
  • 8. G. METAFUNE, A. RHANDI, J. PRÜSS, R. SCHNAUBELT: $L^p$regularity for elliptic operators with unbounded coefficients, preprint.
  • 9. P.RABIER: Elliptic problems on $\mathbb{R} ^N$ with unbounded coefficients in classical Sobolev spaces, Math. Z. 249 (2005), 1-30. MR 2106968
  • 10. O.S. ROTHAUS: Logarithmic Sobolev Inequalities and the Spectrum of Schrödinger operators, J. Funct. Anal. 42 (1981), 110-120. MR 0620582 (83f:58080b)
  • 11. FENG-YU WANG: Logarithmic Sobolev inequalities: conditions and counterexamples, J. Operator Theory 46 (2001), 183-197. MR 1862186 (2003g:58060)

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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: https://doi.org/10.1090/S0002-9939-05-08068-8
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

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