November  2013, 33(11&12): 5457-5491. doi: 10.3934/dcds.2013.33.5457

Hardy type inequalities and hidden energies

1. 

Departamento de Matemáticas and ICMAT. Universidad Autónoma de Madrid, Cantoblanco. 28049 Madrid, Spain

2. 

Department of Mathematics & Engineering Sciences, Hellenic Army Academy, 16673, Athens, Greece

Received  January 2012 Published  May 2013

We obtain new insights into Hardy type Inequalities and the evolution problems associated to them. Surprisingly, the connection of the energy with the Hardy functionals is nontrivial, due to the presence of a Hardy singularity energy. This corresponds to a loss for the total energy. These problems are defined on bounded domains or the whole space.
    We also consider equivalent problems with inverse square potential on exterior domains or the whole space. The extra energy term is then present as an effect that comes from infinity, a kind of hidden energy. In this case, in an unexpected way, this term is additive to the total energy, and it may even constitute the main part of it.
Citation: Juan Luis Vázquez, Nikolaos B. Zographopoulos. Hardy type inequalities and hidden energies. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5457-5491. doi: 10.3934/dcds.2013.33.5457
References:
[1]

Adimurthi, S. Filippas and A. Tertikas, On the best constant of Hardy-Sobolev inequalities,, Nonlinear An. TMA, 70 (2009), 2826. doi: 10.1016/j.na.2008.12.019. Google Scholar

[2]

P. Baras and J. A. Goldstein, The heat equation with a singular potential,, Trans. Amer. Math. Soc., 284 (1984), 121. doi: 10.1090/S0002-9947-1984-0742415-3. Google Scholar

[3]

H. Brezis and M. Marcus, Hardy's inequality revisited,, Ann. Sc. Norm. Pisa, 25 (1997), 217. Google Scholar

[4]

H. Brezis and J. L. Vázquez, Blowup solutions of some nonlinear elliptic problems,, Revista Mat. Univ. Complutense Madrid, 10 (1997), 443. Google Scholar

[5]

X. Cabré and Y. Martel, Existence versus explosion instantané pour des equations de lachaleur linéaires avec potentiel singulier,, C. R. Acad. Sci. Paris, 329 (1999), 973. doi: 10.1016/S0764-4442(00)88588-2. Google Scholar

[6]

C. Cazacu, Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results,, J. Funct. Anal., 263 (2012), 3741. doi: 10.1016/j.jfa.2012.09.006. Google Scholar

[7]

E. B. Davies, A review of Hardy inequalities,, Oper. Theory Adv. Appl., 110 (1999), 55. Google Scholar

[8]

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation,, Non. Anal. TMA, 11 (1987), 1103. doi: 10.1016/0362-546X(87)90001-0. Google Scholar

[9]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities,, J. Funct. Anal., 192 (2002), 186. doi: 10.1006/jfan.2001.3900. Google Scholar

[10]

N. Ghoussoub and A. Moradifam, Bessel potentials and optimal Hardy and Hardy-Rellich inequalities,, Math. Ann., 349 (2011), 1. doi: 10.1007/s00208-010-0510-x. Google Scholar

[11]

J. A. Goldstein and Q. S. Zang, Linear parabolic equations with strong singular potentials,, Trans. Amer. Math. Soc., 355 (2003), 197. doi: 10.1090/S0002-9947-02-03057-X. Google Scholar

[12]

A. Kufner, L. Maligranda and L.-E. Persson, The Hardy inequality. About its history and some related results,, Vydavatelsky' Servis, (2007). Google Scholar

[13]

V. G. Maz'ja, "Sobolev Spaces,", Springer-Verlag, (1985). Google Scholar

[14]

R. Musina, A note on the paper [9],, J. Funct. Anal., 256 (2009), 2741. doi: 10.1016/j.jfa.2008.08.009. Google Scholar

[15]

B. Opic and A. Kufner, Hardy type inequalities,, Pitman Rechearch Notes in Math., 219 (1990). Google Scholar

[16]

J. L. Vázquez and N. B. Zographopoulos, Functional aspects of the Hardy inequality. Appearance of a hidden energy,, J. Evol. Equ., 12 (2012), 713. doi: 10.1007/s00028-012-0151-5. Google Scholar

[17]

J. L. Vázquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential,, J. Funct. Anal., 173 (2000), 103. doi: 10.1006/jfan.1999.3556. Google Scholar

[18]

N. B. Zographopoulos, Existence of extremal functions for a Hardy-Sobolev inequality,, J. Funct. Anal., 259 (2010), 308. doi: 10.1016/j.jfa.2010.03.020. Google Scholar

show all references

References:
[1]

Adimurthi, S. Filippas and A. Tertikas, On the best constant of Hardy-Sobolev inequalities,, Nonlinear An. TMA, 70 (2009), 2826. doi: 10.1016/j.na.2008.12.019. Google Scholar

[2]

P. Baras and J. A. Goldstein, The heat equation with a singular potential,, Trans. Amer. Math. Soc., 284 (1984), 121. doi: 10.1090/S0002-9947-1984-0742415-3. Google Scholar

[3]

H. Brezis and M. Marcus, Hardy's inequality revisited,, Ann. Sc. Norm. Pisa, 25 (1997), 217. Google Scholar

[4]

H. Brezis and J. L. Vázquez, Blowup solutions of some nonlinear elliptic problems,, Revista Mat. Univ. Complutense Madrid, 10 (1997), 443. Google Scholar

[5]

X. Cabré and Y. Martel, Existence versus explosion instantané pour des equations de lachaleur linéaires avec potentiel singulier,, C. R. Acad. Sci. Paris, 329 (1999), 973. doi: 10.1016/S0764-4442(00)88588-2. Google Scholar

[6]

C. Cazacu, Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results,, J. Funct. Anal., 263 (2012), 3741. doi: 10.1016/j.jfa.2012.09.006. Google Scholar

[7]

E. B. Davies, A review of Hardy inequalities,, Oper. Theory Adv. Appl., 110 (1999), 55. Google Scholar

[8]

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation,, Non. Anal. TMA, 11 (1987), 1103. doi: 10.1016/0362-546X(87)90001-0. Google Scholar

[9]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities,, J. Funct. Anal., 192 (2002), 186. doi: 10.1006/jfan.2001.3900. Google Scholar

[10]

N. Ghoussoub and A. Moradifam, Bessel potentials and optimal Hardy and Hardy-Rellich inequalities,, Math. Ann., 349 (2011), 1. doi: 10.1007/s00208-010-0510-x. Google Scholar

[11]

J. A. Goldstein and Q. S. Zang, Linear parabolic equations with strong singular potentials,, Trans. Amer. Math. Soc., 355 (2003), 197. doi: 10.1090/S0002-9947-02-03057-X. Google Scholar

[12]

A. Kufner, L. Maligranda and L.-E. Persson, The Hardy inequality. About its history and some related results,, Vydavatelsky' Servis, (2007). Google Scholar

[13]

V. G. Maz'ja, "Sobolev Spaces,", Springer-Verlag, (1985). Google Scholar

[14]

R. Musina, A note on the paper [9],, J. Funct. Anal., 256 (2009), 2741. doi: 10.1016/j.jfa.2008.08.009. Google Scholar

[15]

B. Opic and A. Kufner, Hardy type inequalities,, Pitman Rechearch Notes in Math., 219 (1990). Google Scholar

[16]

J. L. Vázquez and N. B. Zographopoulos, Functional aspects of the Hardy inequality. Appearance of a hidden energy,, J. Evol. Equ., 12 (2012), 713. doi: 10.1007/s00028-012-0151-5. Google Scholar

[17]

J. L. Vázquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential,, J. Funct. Anal., 173 (2000), 103. doi: 10.1006/jfan.1999.3556. Google Scholar

[18]

N. B. Zographopoulos, Existence of extremal functions for a Hardy-Sobolev inequality,, J. Funct. Anal., 259 (2010), 308. doi: 10.1016/j.jfa.2010.03.020. Google Scholar

[1]

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

[2]

Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations & Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531

[3]

Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations & Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022

[4]

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

[5]

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

[6]

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

[7]

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

[8]

Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 597-608. doi: 10.3934/dcds.2020024

[9]

Tian Ma, Shouhong Wang. Gravitational Field Equations and Theory of Dark Matter and Dark Energy. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 335-366. doi: 10.3934/dcds.2014.34.335

[10]

Yaoping Chen, Jianqing Chen. Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073

[11]

Veronica Felli, Ana Primo. Classification of local asymptotics for solutions to heat equations with inverse-square potentials. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 65-107. doi: 10.3934/dcds.2011.31.65

[12]

Barbara Brandolini, Francesco Chiacchio, Cristina Trombetti. Hardy type inequalities and Gaussian measure. Communications on Pure & Applied Analysis, 2007, 6 (2) : 411-428. doi: 10.3934/cpaa.2007.6.411

[13]

Paola Loreti, Daniela Sforza. Inverse observability inequalities for integrodifferential equations in square domains. Evolution Equations & Control Theory, 2018, 7 (1) : 61-77. doi: 10.3934/eect.2018004

[14]

Angelo Alvino, Roberta Volpicelli, Bruno Volzone. A remark on Hardy type inequalities with remainder terms. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 801-807. doi: 10.3934/dcdss.2011.4.801

[15]

Toshiyuki Suzuki. Nonlinear Schrödinger equations with inverse-square potentials in two dimensional space. Conference Publications, 2015, 2015 (special) : 1019-1024. doi: 10.3934/proc.2015.1019

[16]

Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649

[17]

Eleftherios Gkioulekas, Ka Kit Tung. Is the subdominant part of the energy spectrum due to downscale energy cascade hidden in quasi-geostrophic turbulence?. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 293-314. doi: 10.3934/dcdsb.2007.7.293

[18]

Annalisa Cesaroni, Matteo Novaga. Volume constrained minimizers of the fractional perimeter with a potential energy. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 715-727. doi: 10.3934/dcdss.2017036

[19]

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

[20]

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

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (4)

[Back to Top]