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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

Simona Fornaro 1, , Federica Gregorio 2, and  Abdelaziz Rhandi 3,

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.
Keywords: positivity preserving $C_0$-semigroup, core, unbounded diffusion., Inverse square potential, Hardy's inequality, dissipative and dispersive operator.
Mathematics Subject Classification: Primary: 47D06, 35P05, 35J70; Secondary: 35K65, 34G1.
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 & Applied Analysis, 2016, 15 (6) : 2357-2372. doi: 10.3934/cpaa.2016040
References:
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A. Canale, A. Rhandi and C. Tacelli, Schrödinger type operators with unbounded diffusion and potential terms,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, ().   Google Scholar

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show all references

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.}, ().   Google Scholar

[2]

Trans. Am. Math. Soc., 284 (1984), 121-139. doi: 10.2307/1999277.  Google Scholar

[3]

Discrete Cont. Dyn. Syst. S., 6 (2013), 649-655.  Google Scholar

[4]

Clarendon Press, Oxford, 1987.  Google Scholar

[5]

Springer-Verlag, New York, 2000.  Google Scholar

[6]

Discrete Contin. Dyn. Syst., 18 (2007), 747-772. doi: 10.3934/dcds.2007.18.747.  Google Scholar

[7]

Discrete Contin. Dyn. Syst., 33 (2013), 5049-5058.  Google Scholar

[8]

Springer, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[9]

J. Evol. Equ., 15 (2015), 53-88. doi: 10.1007/s00028-014-0249-z.  Google Scholar

[10]

Mediterranean Journal of Mathematics, 5 (2008), 357-369. doi: 10.1007/s00009-008-0155-0.  Google Scholar

[11]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., XI (2012), 303-340.  Google Scholar

[12]

J. Evol. Equ., 16 (2016), 391-439. doi: 10.1007/s00028-015-0307-1.  Google Scholar

[13]

Mat. Zametki, 67 (2000), 563-572. doi: 10.1007/BF02676404.  Google Scholar

[14]

Lecture Notes in Math. 1184, Springer-Verlag, 1986. doi: 10.1007/BFb0074922.  Google Scholar

[15]

J. Math. Soc. Japan, 34 (1982), 677-701. doi: 10.2969/jmsj/03440677.  Google Scholar

[16]

Japan. J. Math., 22 (1996), 199-239.  Google Scholar

[17]

London Math. Soc. Monographs 31, Princeton Univ. Press 2004.  Google Scholar

[18]

Academic Press, New York, 1975.  Google Scholar

[19]

Arch. Rational Mech. Anal., 52 (1973), 44-48.  Google Scholar

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