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

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.
Citation: Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete and 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.

[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.

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

show all references

References:
[1]

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

[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.

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

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