## Dirichlet boundary conditions for elliptic operators with unbounded drift

HTML articles powered by AMS MathViewer

- by A. Lunardi, G. Metafune and D. Pallara PDF
- Proc. Amer. Math. Soc.
**133**(2005), 2625-2635 Request permission

Erratum: Proc. Amer. Math. Soc.

**134**(2006), 2479-2480.

## 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 $L^2(\Omega , \mu )$.## References

- M. Bertoldi and S. Fornaro,
*Gradient estimates in parabolic problems with unbounded coefficients*, Studia Math.**165**(2004), no. 3, 221–254. MR**2109509**, DOI 10.4064/sm165-3-3 - Sandra Cerrai,
*Second order PDE’s in finite and infinite dimension*, Lecture Notes in Mathematics, vol. 1762, Springer-Verlag, Berlin, 2001. A probabilistic approach. MR**1840644**, DOI 10.1007/b80743 - Giuseppe Da Prato and Alessandra Lunardi,
*Elliptic operators with unbounded drift coefficients and Neumann boundary condition*, J. Differential Equations**198**(2004), no. 1, 35–52. MR**2037749**, DOI 10.1016/j.jde.2003.10.025 - E. B. Davies,
*Heat kernels and spectral theory*, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR**990239**, DOI 10.1017/CBO9780511566158 - Leonard Gross,
*Logarithmic Sobolev inequalities*, Amer. J. Math.**97**(1975), no. 4, 1061–1083. MR**420249**, DOI 10.2307/2373688 - Leonard Gross,
*Logarithmic Sobolev inequalities and contractivity properties of semigroups*, Dirichlet forms (Varenna, 1992) Lecture Notes in Math., vol. 1563, Springer, Berlin, 1993, pp. 54–88. MR**1292277**, DOI 10.1007/BFb0074091 - 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. - G. Metafune, A. Rhandi, J. Prüss, R. Schnaubelt:
*$L^p$ regularity for elliptic operators with unbounded coefficients*, preprint. - Patrick J. Rabier,
*Elliptic problems on $\Bbb R^N$ with unbounded coefficients in classical Sobolev spaces*, Math. Z.**249**(2005), no. 1, 1–30. MR**2106968**, DOI 10.1007/s00209-004-0686-4 - O. S. Rothaus,
*Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities*, J. Functional Analysis**42**(1981), no. 1, 102–109. MR**620581**, DOI 10.1016/0022-1236(81)90049-5 - Feng-Yu Wang,
*Logarithmic Sobolev inequalities: conditions and counterexamples*, J. Operator Theory**46**(2001), no. 1, 183–197. MR**1862186**

## 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
- Received by editor(s): April 19, 2004
- Published electronically: April 19, 2005
- Communicated by: David S. Tartakoff
- © Copyright 2005 American Mathematical Society
- 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
- MathSciNet review: 2146208