American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food
  • CPAA Home
  • This Issue
  • Next Article
    New general decay result for a system of viscoelastic wave equations with past history
January  2021, 20(1): 405-425. doi: 10.3934/cpaa.2020274

Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials

Anna Canale 1,, , Francesco Pappalardo 2, and  Ciro Tarantino 3,

1. 

Dipartimento di Ingegneria dell'Informazione ed Elettrica e Matematica Applicata, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, SA, Italy
2. 

Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli Federico Ⅱ, Complesso Universitario Monte S. Angelo, Via Cintia, 80126 Naples, Italy
3. 

Dipartimento di Scienze Economiche e Statistiche, Università degli Studi di Napoli Federico Ⅱ, Complesso Universitario Monte S. Angelo, Via Cintia, 80126 Naples, Italy

* Corresponding author

Received  March 2020 Revised  September 2020 Published  January 2021 Early access  November 2020

Fund Project: The first two authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

The main results in the paper are the weighted multipolar Hardy inequalities
$ \begin{equation*} c\int_{\mathbb{R}^N}\sum\limits_{i = 1}^n\frac{\varphi^2}{|x-a_i|^2}\,\mu(x)dx \leq\int_{\mathbb{R}^N}|\nabla \varphi |^2\mu(x)dx+ K\int_{\mathbb{R}^N} \varphi^2\mu(x)dx, \end{equation*} $
in
$ \mathbb{R}^N $
 for any
$ \varphi $
 in a suitable weighted Sobolev space, with
$ 0<c\le c_{o,\mu} $
,
$ a_1,\dots,a_n\in \mathbb{R}^N $
,
$ K $
 constant. The weight functions
$ \mu $
 are of a quite general type.
The paper fits in the framework of Kolmogorov operators defined on smooth functions
$ \begin{equation*} Lu = \Delta u+\frac{\nabla \mu}{\mu}\cdot\nabla u, \end{equation*} $
perturbed by multipolar inverse square potentials, and related evolution problems. Necessary and sufficient conditions for the existence of exponentially bounded in time positive solutions to the associated initial value problem are based on weighted Hardy inequalities. For constants
$ c $
 beyond the optimal Hardy constant
$ c_{o,\mu} $
 we are able to show nonexistence of positive solutions.
Keywords: Weighted Hardy inequality, optimal constant, Kolmogorov operators, multipolar potentials, existence and nonexistence results.
Mathematics Subject Classification: Primary: 35K15, 35K65, 35B25; Secondary: 34G10, 47D03.
Citation: Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure and Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274
References:
[1]

A. Albanese, L. Lorenzi and E. Mangino, $L^p$–uniqueness for elliptic operators with unbounded coefficients in $\mathbb{R} ^N$, J. Funct. Anal., 256 (2009), 1238-1257.  doi: 10.1016/j.jfa.2008.07.022.

[2]

D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Sc. Norm. Super. Pisa, 22 (1968), 607-694. 

[3]

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.

[4]

R. Bosi, J. Dolbeault and M. J. Esteban, Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators, Commun. Pure Appl. Anal., 7 (2008), 533-562.  doi: 10.3934/cpaa.2008.7.533.

[5]

X. Cabré and Y. Martel, Existence versus explosion instantanée pour des e$\acute{\rm{q}}$uations de la chaleur lineáires avec potentiel singulier, C. R. Acad. Sci. Paris, 329 (1999), 973-978.  doi: 10.1016/S0764-4442(00)88588-2.

[6]

A. Canale, F. Gregorio, A. Rhandi and C. Tacelli, Weighted Hardy's inequalities and Kolmogorov-type operators, Appl. Anal., 98 (2019), 1236-1254.  doi: 10.1080/00036811.2017.1419200.

[7]

A. Canale, R. M. Mininni and A. Rhandi, Analytic approach to solve a degenerate parabolic PDE for the Heston model, Math. Meth. Appl. Sci., 40 (2017), 4982-4992.  doi: 10.1002/mma.4363.

[8]

A. Canale and F. Pappalardo, Weighted Hardy inequalities and Ornstein-Uhlenbeck type operators perturbed by multipolar inverse square potentials, J. Math. Anal. Appl., 463 (2018), 895-909.  doi: 10.1016/j.jmaa.2018.03.059.

[9]

A. Canale, F. Pappalardo and C. Tarantino, A class of weighted Hardy inequalities and applications to evolution problems, Ann. Mat. Pura Appl., 199 (2020), 1171-1181.  doi: 10.1007/s10231-019-00916-y.

[10]

A. Canale, A. Rhandi and C. Tacelli, Schrödinger type operators with unbounded diffusion and potential terms, Ann. Sc. Norm. Super. Pisa Cl. Sci., XVI (2016), 581-601.  doi: 10.2422/2036-2145.201409_007.

[11]

A. Canale, A. Rhandi and C. Tacelli, Kernel estimates for Schrödinger type operators with unbounded diffusion and potential terms, Z. Anal. Anwend., 36 (2017), 377-392.  doi: 10.4171/ZAA/1593.

[12]

A. Canale and C. Tacelli, Kernel estimates for a Schrödinger type operator, Riv. Mat. Univ. Parma, 7 (2016), 341-350. 

[13]

C. Cazacu, New estimates for the Hardy constants of multipolar Schrödinger operators, Commun. Contemp. Math., 18 (2016), 1-28.  doi: 10.1142/S0219199715500935.

[14]

C. Cazacu and E. Zuazua, Improved multipolar Hardy inequalities, in Studies in Phase Space Analysis of PDEs (eds. M. Cicognani, F. Colombini and D. Del Santo), Progress in Nonlinear Differential Equations and Their Applications 84, Birkhäuser, New York (2013), 37–52. doi: 10.1007/978-1-4614-6348-1_3.

[15]

V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multipolar inverse-square potentials, J. Funct. Anal., 250 (2007), 265-316.  doi: 10.1016/j.jfa.2006.10.019.

[16]

G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential, Appl. Anal., 91 (2012), 2057-2071.  doi: 10.1080/00036811.2011.587809.

[17]

O. Ladyz'enskaya, V. Solonnikov and N. Ural'tseva, Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, Rhode Island, 1968.

[18]

L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups, Pure and Applied Mathematics, CRC Press, 2006.

[19]

E. Mitidieri, A simple approach to Hardy inequalities, Math. Notes, 67 (2000), 479-486.  doi: 10.1007/BF02676404.

[20]

J. D. Morgan, Schrödinger operators whose potentials have separated singularities, J. Operat. Theor., 1 (1979), 109-115. 

[21]

B. Simon, Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions, Ann. Inst. H. Poincaré Sect. A (N.S.), 38 (1983), 295-308. 

[22]

J. M. Tölle, Uniqueness of weighted Sobolev spaces with weakly differentiable weights, J. Funct. Anal., 263 (2012), 3195-3223.  doi: 10.1016/j.jfa.2012.08.002.

show all references

References:
[1]

A. Albanese, L. Lorenzi and E. Mangino, $L^p$–uniqueness for elliptic operators with unbounded coefficients in $\mathbb{R} ^N$, J. Funct. Anal., 256 (2009), 1238-1257.  doi: 10.1016/j.jfa.2008.07.022.

[2]

D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Sc. Norm. Super. Pisa, 22 (1968), 607-694. 

[3]

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.

[4]

R. Bosi, J. Dolbeault and M. J. Esteban, Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators, Commun. Pure Appl. Anal., 7 (2008), 533-562.  doi: 10.3934/cpaa.2008.7.533.

[5]

X. Cabré and Y. Martel, Existence versus explosion instantanée pour des e$\acute{\rm{q}}$uations de la chaleur lineáires avec potentiel singulier, C. R. Acad. Sci. Paris, 329 (1999), 973-978.  doi: 10.1016/S0764-4442(00)88588-2.

[6]

A. Canale, F. Gregorio, A. Rhandi and C. Tacelli, Weighted Hardy's inequalities and Kolmogorov-type operators, Appl. Anal., 98 (2019), 1236-1254.  doi: 10.1080/00036811.2017.1419200.

[7]

A. Canale, R. M. Mininni and A. Rhandi, Analytic approach to solve a degenerate parabolic PDE for the Heston model, Math. Meth. Appl. Sci., 40 (2017), 4982-4992.  doi: 10.1002/mma.4363.

[8]

A. Canale and F. Pappalardo, Weighted Hardy inequalities and Ornstein-Uhlenbeck type operators perturbed by multipolar inverse square potentials, J. Math. Anal. Appl., 463 (2018), 895-909.  doi: 10.1016/j.jmaa.2018.03.059.

[9]

A. Canale, F. Pappalardo and C. Tarantino, A class of weighted Hardy inequalities and applications to evolution problems, Ann. Mat. Pura Appl., 199 (2020), 1171-1181.  doi: 10.1007/s10231-019-00916-y.

[10]

A. Canale, A. Rhandi and C. Tacelli, Schrödinger type operators with unbounded diffusion and potential terms, Ann. Sc. Norm. Super. Pisa Cl. Sci., XVI (2016), 581-601.  doi: 10.2422/2036-2145.201409_007.

[11]

A. Canale, A. Rhandi and C. Tacelli, Kernel estimates for Schrödinger type operators with unbounded diffusion and potential terms, Z. Anal. Anwend., 36 (2017), 377-392.  doi: 10.4171/ZAA/1593.

[12]

A. Canale and C. Tacelli, Kernel estimates for a Schrödinger type operator, Riv. Mat. Univ. Parma, 7 (2016), 341-350. 

[13]

C. Cazacu, New estimates for the Hardy constants of multipolar Schrödinger operators, Commun. Contemp. Math., 18 (2016), 1-28.  doi: 10.1142/S0219199715500935.

[14]

C. Cazacu and E. Zuazua, Improved multipolar Hardy inequalities, in Studies in Phase Space Analysis of PDEs (eds. M. Cicognani, F. Colombini and D. Del Santo), Progress in Nonlinear Differential Equations and Their Applications 84, Birkhäuser, New York (2013), 37–52. doi: 10.1007/978-1-4614-6348-1_3.

[15]

V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multipolar inverse-square potentials, J. Funct. Anal., 250 (2007), 265-316.  doi: 10.1016/j.jfa.2006.10.019.

[16]

G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential, Appl. Anal., 91 (2012), 2057-2071.  doi: 10.1080/00036811.2011.587809.

[17]

O. Ladyz'enskaya, V. Solonnikov and N. Ural'tseva, Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, Rhode Island, 1968.

[18]

L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups, Pure and Applied Mathematics, CRC Press, 2006.

[19]

E. Mitidieri, A simple approach to Hardy inequalities, Math. Notes, 67 (2000), 479-486.  doi: 10.1007/BF02676404.

[20]

J. D. Morgan, Schrödinger operators whose potentials have separated singularities, J. Operat. Theor., 1 (1979), 109-115. 

[21]

B. Simon, Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions, Ann. Inst. H. Poincaré Sect. A (N.S.), 38 (1983), 295-308. 

[22]

J. M. Tölle, Uniqueness of weighted Sobolev spaces with weakly differentiable weights, J. Funct. Anal., 263 (2012), 3195-3223.  doi: 10.1016/j.jfa.2012.08.002.

[1]

Roberta Bosi, Jean Dolbeault, Maria J. Esteban. Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators. Communications on Pure and Applied Analysis, 2008, 7 (3) : 533-562. doi: 10.3934/cpaa.2008.7.533
[2]

Qianzhong Ou. Nonexistence results for a fully nonlinear evolution inequality. Electronic Research Announcements, 2016, 23: 19-24. doi: 10.3934/era.2016.23.003
[3]

Craig Cowan. Optimal Hardy inequalities for general elliptic operators with improvements. Communications on Pure and Applied Analysis, 2010, 9 (1) : 109-140. doi: 10.3934/cpaa.2010.9.109
[4]

Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121
[5]

Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935
[6]

Ewa Schmeidel, Karol Gajda, Tomasz Gronek. On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2681-2690. doi: 10.3934/dcdsb.2014.19.2681
[7]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[8]

B. Abdellaoui, E. Colorado, I. Peral. Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions. Communications on Pure and Applied Analysis, 2006, 5 (1) : 29-54. doi: 10.3934/cpaa.2006.5.29
[9]

Xiao-Jing Zhong, Chun-Lei Tang. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Communications on Pure and Applied Analysis, 2017, 16 (2) : 611-628. doi: 10.3934/cpaa.2017030
[10]

Alexei Shadrin. The Landau--Kolmogorov inequality revisited. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1183-1210. doi: 10.3934/dcds.2014.34.1183
[11]

Rémi Carles. Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 385-398. doi: 10.3934/dcds.2005.13.385
[12]

Lucio Boccardo, Luigi Orsina, Ireneo Peral. A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 513-523. doi: 10.3934/dcds.2006.16.513
[13]

Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018
[14]

Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143
[15]

Kods Hassine. Existence and uniqueness of radial solutions for Hardy-Hénon equations involving k-Hessian operators. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022084
[16]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1945-1966. doi: 10.3934/dcdss.2020469
[17]

Peter Boyvalenkov, Maya Stoyanova. New nonexistence results for spherical designs. Advances in Mathematics of Communications, 2013, 7 (3) : 279-292. doi: 10.3934/amc.2013.7.279
[18]

Ana-Maria Acu, Laura Hodis, Ioan Rasa. Multivariate weighted kantorovich operators. Mathematical Foundations of Computing, 2020, 3 (2) : 117-124. doi: 10.3934/mfc.2020009
[19]

Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure and Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011
[20]

Ryan Loxton, Qun Lin, Volker Rehbock, Kok Lay Teo. Control parameterization for optimal control problems with continuous inequality constraints: New convergence results. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 571-599. doi: 10.3934/naco.2012.2.571

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (168)
  • HTML views (48)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy