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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

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.
Citation: Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274
References:
[1]

A. AlbaneseL. 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

R. BosiJ. 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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

A. CanaleR. 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.  Google Scholar

[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.  Google Scholar

[9]

A. CanaleF. 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.  Google Scholar

[10]

A. CanaleA. 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.  Google Scholar

[11]

A. CanaleA. 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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[15]

V. FelliE. 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.  Google Scholar

[16]

G. R. GoldsteinJ. 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.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[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.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. AlbaneseL. 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

R. BosiJ. 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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

A. CanaleR. 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.  Google Scholar

[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.  Google Scholar

[9]

A. CanaleF. 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.  Google Scholar

[10]

A. CanaleA. 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.  Google Scholar

[11]

A. CanaleA. 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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[15]

V. FelliE. 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.  Google Scholar

[16]

G. R. GoldsteinJ. 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.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[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.   Google Scholar

[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.  Google Scholar

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