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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) |
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. |
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.}, ().
|
[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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