American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for generalized Chaplygin gas
  • CPAA Home
  • This Issue
  • Next Article
    Steady state solutions of ferrofluid flow models
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

[1]

Jiří Neustupa. On $L^2$-Boundedness of a $C_0$-Semigroup generated by the perturbed oseen-type operator arising from flow around a rotating body. Conference Publications, 2007, 2007 (Special) : 758-767. doi: 10.3934/proc.2007.2007.758
[2]

Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361
[3]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387
[4]

Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577
[5]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
[6]

José A. Conejero, Alfredo Peris. Hypercyclic translation $C_0$-semigroups on complex sectors. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1195-1208. doi: 10.3934/dcds.2009.25.1195
[7]

Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143
[8]

Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97
[9]

Gisèle Ruiz Goldstein, Jerome A. Goldstein, Abdelaziz Rhandi. Kolmogorov equations perturbed by an inverse-square potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 623-630. doi: 10.3934/dcdss.2011.4.623
[10]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623
[11]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427
[12]

Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195
[13]

Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176
[14]

Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, Jiqiang Zheng. The energy-critical NLS with inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3831-3866. doi: 10.3934/dcds.2017162
[15]

Patrick Martinez, Judith Vancostenoble. The cost of boundary controllability for a parabolic equation with inverse square potential. Evolution Equations & Control Theory, 2019, 8 (2) : 397-422. doi: 10.3934/eect.2019020
[16]

Adam Bobrowski, Adam Gregosiewicz, Małgorzata Murat. Functionals-preserving cosine families generated by Laplace operators in C[0,1]. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1877-1895. doi: 10.3934/dcdsb.2015.20.1877
[17]

Sallah Eddine Boutiah, Abdelaziz Rhandi, Cristian Tacelli. Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 803-817. doi: 10.3934/dcds.2019033
[18]

Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541
[19]

Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074
[20]

Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control & Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences