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November  2013, 33(11&12): 5049-5058. doi: 10.3934/dcds.2013.33.5049

On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces

Simona Fornaro 1, and  Abdelaziz Rhandi 2,

1. 

Università degli Studi di Pavia, Dipartimento di Matematica “F. Casorati”, via Ferrata 1, 27100 Pavia
2. 

Dipartimento di Ingegneria dell'Informazione, Ingegneria Elletrica e Matematica Applicata, Università degli Studi di Salerno, Via Ponte Don Melillo, 84084 Fisciano (Sa)

Received  November 2011 Published  May 2013

In this paper we give sufficient conditions ensuring that the space of test functions $C_c^{\infty}(R^N)$ is a core for the operator $$L_0u=\Delta u-Mx\cdot \nabla u+\frac{\alpha}{|x|^2}u=:Lu+\frac{\alpha}{|x|^2}u,$$ and $L_0$ with domain $W_\mu^{2,p}(R^N)$ generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p_\mu(R^N),\,1 < p < \infty$. Here $M$ is a positive definite $N\times N$ hermitian matrix and $\mu$ is the unique invariant measure for the Ornstein-Uhlenbeck operator $L$. The proofs are based on an $L^p$-weighted Hardy's inequality and perturbation techniques.
Keywords: Ornstein-Uhlenbeck operator., core, dissipative and dispersive operator, Hardy's inequality, Inverse square potential, positivity preserving $C_0$-semigroup.
Mathematics Subject Classification: Primary: 47D06, 35R05, 35B25; Secondary: 35K15, 35K65, 34G1.
Citation: Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049
References:
[1]

M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," Chapman Hall/CRC, Boca Raton FL, 2007.  Google Scholar

[2]

T. Durante and A. Rhandi, On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials, Discrete Cont. Dyn. Syst. S., 6 (2013), 649-655. doi: doi:10.3934/dcdss.2013.6.649.  Google Scholar

[3]

D. E. Edmunds and W. E. Evans, "Spectral Theory and Differential Operators," Clarendon Press, Oxford, 1987.  Google Scholar

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K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 2000.  Google Scholar

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G. R. Goldstein, J. A. Goldstein and A. Rhandi, Kolmogorov equation perturbed by an inverse-square potential, Discrete Cont. Dyn. Syst. S., 4 (2011), 623-630. doi: 10.3934/dcdss.2011.4.623.  Google Scholar

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G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential, Applicable Analysis, 91 (2012), 2057-2071. doi: 10.1080/00036811.2011.587809.  Google Scholar

[7]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, 1983.  Google Scholar

[8]

G. Metafune, J. Prüss, A. Rhandi and R. Schnaubelt, The domain of the Ornstein-Uhlenbeck operator on an $L^p$-space with invariant measure, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), I (2002), 471-485.  Google Scholar

[9]

R. Nagel, "One-Parameter Semigroups of Positive Operators," Lecture Notes in Math., 1184, Springer-Verlag, 1986.  Google Scholar

[10]

N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan, 34 (1982), 677-701. doi: 10.2969/jmsj/03440677.  Google Scholar

[11]

N. Okazawa, $L^p$-theory of Schrödinger operators with strongly singular potentials, Japan. J. Math., 22 (1996), 199-239.  Google Scholar

[12]

E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Math. Soc. Monographs, 31. Princeton Univ. Press 2004.  Google Scholar

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M. Reed and B. Simon, "Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness," Academic Press, New York, 1975.  Google Scholar

[14]

B. Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Anal., 52 (1973), 44-48.  Google Scholar

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J. Walter, Note on a paper by Stetkœr-Hansen concerning essential self-adjointness of Schrödinger operators, Math. Scand., 25 (1969), 94-96.  Google Scholar

show all references

References:
[1]

M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," Chapman Hall/CRC, Boca Raton FL, 2007.  Google Scholar

[2]

T. Durante and A. Rhandi, On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials, Discrete Cont. Dyn. Syst. S., 6 (2013), 649-655. doi: doi:10.3934/dcdss.2013.6.649.  Google Scholar

[3]

D. E. Edmunds and W. E. Evans, "Spectral Theory and Differential Operators," Clarendon Press, Oxford, 1987.  Google Scholar

[4]

K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 2000.  Google Scholar

[5]

G. R. Goldstein, J. A. Goldstein and A. Rhandi, Kolmogorov equation perturbed by an inverse-square potential, Discrete Cont. Dyn. Syst. S., 4 (2011), 623-630. doi: 10.3934/dcdss.2011.4.623.  Google Scholar

[6]

G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential, Applicable Analysis, 91 (2012), 2057-2071. doi: 10.1080/00036811.2011.587809.  Google Scholar

[7]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, 1983.  Google Scholar

[8]

G. Metafune, J. Prüss, A. Rhandi and R. Schnaubelt, The domain of the Ornstein-Uhlenbeck operator on an $L^p$-space with invariant measure, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), I (2002), 471-485.  Google Scholar

[9]

R. Nagel, "One-Parameter Semigroups of Positive Operators," Lecture Notes in Math., 1184, Springer-Verlag, 1986.  Google Scholar

[10]

N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan, 34 (1982), 677-701. doi: 10.2969/jmsj/03440677.  Google Scholar

[11]

N. Okazawa, $L^p$-theory of Schrödinger operators with strongly singular potentials, Japan. J. Math., 22 (1996), 199-239.  Google Scholar

[12]

E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Math. Soc. Monographs, 31. Princeton Univ. Press 2004.  Google Scholar

[13]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness," Academic Press, New York, 1975.  Google Scholar

[14]

B. Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Anal., 52 (1973), 44-48.  Google Scholar

[15]

J. Walter, Note on a paper by Stetkœr-Hansen concerning essential self-adjointness of Schrödinger operators, Math. Scand., 25 (1969), 94-96.  Google Scholar

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