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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:
[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]

P. Baras and J. A. Goldstein, The heat equation with a singular potential,, \emph{Trans. Am. Math. Soc.}, 284 (1984), 121. doi: 10.2307/1999277. Google Scholar

[3]

T. Durante and A. Rhandi, On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials,, \emph{Discrete Cont. Dyn. Syst. S.}, 6 (2013), 649. Google Scholar

[4]

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

[5]

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

[6]

S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces,, \emph{Discrete Contin. Dyn. Syst.}, 18 (2007), 747. doi: 10.3934/dcds.2007.18.747. Google Scholar

[7]

S. Fornaro and A. Rhandi, On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$-spaces,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 5049. Google Scholar

[8]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer, (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar

[9]

L. Lorenzi and A. Rhandi, On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates,, \emph{J. Evol. Equ.}, 15 (2015), 53. doi: 10.1007/s00028-014-0249-z. Google Scholar

[10]

G. Metafune and C. Spina, An integration by parts formula in Sobolev spaces,, \emph{Mediterranean Journal of Mathematics}, 5 (2008), 357. doi: 10.1007/s00009-008-0155-0. Google Scholar

[11]

G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, XI (2012), 303. Google Scholar

[12]

G. Metafune, N. Okazawa, M. Sobajima and C. Spina, Scale invariant elliptic operators with singular coefficients,, \emph{J. Evol. Equ.}, 16 (2016), 391. doi: 10.1007/s00028-015-0307-1. Google Scholar

[13]

E. Mitidieri, A simple approach to Hardy inequalities,, \emph{Mat. Zametki}, 67 (2000), 563. doi: 10.1007/BF02676404. Google Scholar

[14]

R. Nagel (ed.), One-Parameter Semigroups of Positive Operators,, Lecture Notes in Math. \textbf{1184}, 1184 (1986). doi: 10.1007/BFb0074922. Google Scholar

[15]

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

[16]

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

[17]

E. M. Ouhabaz, Analysis of Heat Equations on Domains,, London Math. Soc. Monographs \textbf{31}, 31 (2004). Google Scholar

[18]

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

[19]

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

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]

P. Baras and J. A. Goldstein, The heat equation with a singular potential,, \emph{Trans. Am. Math. Soc.}, 284 (1984), 121. doi: 10.2307/1999277. Google Scholar

[3]

T. Durante and A. Rhandi, On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials,, \emph{Discrete Cont. Dyn. Syst. S.}, 6 (2013), 649. Google Scholar

[4]

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

[5]

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

[6]

S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces,, \emph{Discrete Contin. Dyn. Syst.}, 18 (2007), 747. doi: 10.3934/dcds.2007.18.747. Google Scholar

[7]

S. Fornaro and A. Rhandi, On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$-spaces,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 5049. Google Scholar

[8]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer, (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar

[9]

L. Lorenzi and A. Rhandi, On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates,, \emph{J. Evol. Equ.}, 15 (2015), 53. doi: 10.1007/s00028-014-0249-z. Google Scholar

[10]

G. Metafune and C. Spina, An integration by parts formula in Sobolev spaces,, \emph{Mediterranean Journal of Mathematics}, 5 (2008), 357. doi: 10.1007/s00009-008-0155-0. Google Scholar

[11]

G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, XI (2012), 303. Google Scholar

[12]

G. Metafune, N. Okazawa, M. Sobajima and C. Spina, Scale invariant elliptic operators with singular coefficients,, \emph{J. Evol. Equ.}, 16 (2016), 391. doi: 10.1007/s00028-015-0307-1. Google Scholar

[13]

E. Mitidieri, A simple approach to Hardy inequalities,, \emph{Mat. Zametki}, 67 (2000), 563. doi: 10.1007/BF02676404. Google Scholar

[14]

R. Nagel (ed.), One-Parameter Semigroups of Positive Operators,, Lecture Notes in Math. \textbf{1184}, 1184 (1986). doi: 10.1007/BFb0074922. Google Scholar

[15]

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

[16]

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

[17]

E. M. Ouhabaz, Analysis of Heat Equations on Domains,, London Math. Soc. Monographs \textbf{31}, 31 (2004). Google Scholar

[18]

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

[19]

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

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