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November  2016, 15(6): 2357-2372. doi: 10.3934/cpaa.2016040

## Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces

 1 Università degli Studi di Pavia, Dipartimento di Matematica “F. Casorati”, via Ferrata 1, 27100 Pavia 2 Dipartimento di Fisica, Universitá degli Studi di Salerno, via Giovanni Paolo II, 132, 84084, Fisciano (Sa), Italy 3 Dipartimento di Ingegneria dell'Informazione e Matematica Applicata, Università degli Studi di Salerno, Via Ponte Don Melillo, 84084 Fisciano (Sa)

Received  March 2016 Revised  June 2016 Published  September 2016

In this paper we give sufficient conditions on $\alpha \ge 0$ and $c\in R$ ensuring that the space of test functions $C_c^\infty(R^N)$ is a core for the operator \begin{eqnarray} L_0u=(1+|x|^\alpha )\Delta u+\frac{c}{|x|^2}u=:Lu+\frac{c}{|x|^2}u, \end{eqnarray} and $L_0$ with a suitable domain generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p(R^N), 1 < p < \infty$. The proofs are based on some $L^p$-weighted Hardy's inequality and perturbation techniques.
Citation: Simona Fornaro, Federica Gregorio, Abdelaziz Rhandi. Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2357-2372. doi: 10.3934/cpaa.2016040
##### References:
 [1] A. Canale, A. Rhandi and C. Tacelli, Schrödinger type operators with unbounded diffusion and potential terms,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, (). [2] P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Am. Math. Soc., 284 (1984), 121-139. doi: 10.2307/1999277. [3] 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. [4] D. E. Edmunds and W. E. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987. [5] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [6] S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Discrete Contin. Dyn. Syst., 18 (2007), 747-772. doi: 10.3934/dcds.2007.18.747. [7] S. Fornaro and A. Rhandi, On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$-spaces, Discrete Contin. Dyn. Syst., 33 (2013), 5049-5058. [8] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. doi: 10.1007/978-3-642-61798-0. [9] L. Lorenzi and A. Rhandi, On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates, J. Evol. Equ., 15 (2015), 53-88. doi: 10.1007/s00028-014-0249-z. [10] G. Metafune and C. Spina, An integration by parts formula in Sobolev spaces, Mediterranean Journal of Mathematics, 5 (2008), 357-369. doi: 10.1007/s00009-008-0155-0. [11] G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci., XI (2012), 303-340. [12] G. Metafune, N. Okazawa, M. Sobajima and C. Spina, Scale invariant elliptic operators with singular coefficients, J. Evol. Equ., 16 (2016), 391-439. doi: 10.1007/s00028-015-0307-1. [13] E. Mitidieri, A simple approach to Hardy inequalities, Mat. Zametki, 67 (2000), 563-572. doi: 10.1007/BF02676404. [14] R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer-Verlag, 1986. doi: 10.1007/BFb0074922. [15] 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. [16] N. Okazawa, $L^p$-theory of Schrödinger operators with strongly singular potentials, Japan. J. Math., 22 (1996), 199-239. [17] E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Math. Soc. Monographs 31, Princeton Univ. Press 2004. [18] M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. [19] B. Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Anal., 52 (1973), 44-48.

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##### References:
 [1] A. Canale, A. Rhandi and C. Tacelli, Schrödinger type operators with unbounded diffusion and potential terms,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, (). [2] P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Am. Math. Soc., 284 (1984), 121-139. doi: 10.2307/1999277. [3] 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. [4] D. E. Edmunds and W. E. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987. [5] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [6] S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Discrete Contin. Dyn. Syst., 18 (2007), 747-772. doi: 10.3934/dcds.2007.18.747. [7] S. Fornaro and A. Rhandi, On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$-spaces, Discrete Contin. Dyn. Syst., 33 (2013), 5049-5058. [8] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. doi: 10.1007/978-3-642-61798-0. [9] L. Lorenzi and A. Rhandi, On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates, J. Evol. Equ., 15 (2015), 53-88. doi: 10.1007/s00028-014-0249-z. [10] G. Metafune and C. Spina, An integration by parts formula in Sobolev spaces, Mediterranean Journal of Mathematics, 5 (2008), 357-369. doi: 10.1007/s00009-008-0155-0. [11] G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci., XI (2012), 303-340. [12] G. Metafune, N. Okazawa, M. Sobajima and C. Spina, Scale invariant elliptic operators with singular coefficients, J. Evol. Equ., 16 (2016), 391-439. doi: 10.1007/s00028-015-0307-1. [13] E. Mitidieri, A simple approach to Hardy inequalities, Mat. Zametki, 67 (2000), 563-572. doi: 10.1007/BF02676404. [14] R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer-Verlag, 1986. doi: 10.1007/BFb0074922. [15] 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. [16] N. Okazawa, $L^p$-theory of Schrödinger operators with strongly singular potentials, Japan. J. Math., 22 (1996), 199-239. [17] E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Math. Soc. Monographs 31, Princeton Univ. Press 2004. [18] M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. [19] B. Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Anal., 52 (1973), 44-48.
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