American Institute of Mathematical Sciences

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

Hardy type inequalities and hidden energies

Juan Luis Vázquez 1, and  Nikolaos B. Zographopoulos 2,

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.
Keywords: Hardy type inequalities, energy at singularity, hidden or dark energy., inverse-square potential.
Mathematics Subject Classification: Primary: 35A23, 46E35; Secondary: 35J20, 35J7.
Citation: Juan Luis Vázquez, Nikolaos B. Zographopoulos. Hardy type inequalities and hidden energies. Discrete & Continuous Dynamical Systems, 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), 2826-2833. 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), 121-139. 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), 217-237.  Google Scholar

[4]

H. Brezis and J. L. Vázquez, Blowup solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense Madrid, 10 (1997), 443-469.  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), 973-978. doi: 10.1016/S0764-4442(00)88588-2.  Google Scholar

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C. Cazacu, Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results, J. Funct. Anal., 263 (2012), 3741-3783. 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), 55-67.  Google Scholar

[8]

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Non. Anal. TMA, 11 (1987), 1103-1133. 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), 186-233; Corrigendum, J. Funct. Anal., 255 (2008), 2095. 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), 1-57. 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), 197-211. 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, Plzen, 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), 2741-2745. 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 Longman (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), 713-739. 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), 103-153. 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), 308-314. 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), 2826-2833. 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), 121-139. 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), 217-237.  Google Scholar

[4]

H. Brezis and J. L. Vázquez, Blowup solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense Madrid, 10 (1997), 443-469.  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), 973-978. 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), 3741-3783. 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), 55-67.  Google Scholar

[8]

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Non. Anal. TMA, 11 (1987), 1103-1133. 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), 186-233; Corrigendum, J. Funct. Anal., 255 (2008), 2095. 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), 1-57. 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), 197-211. 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, Plzen, 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), 2741-2745. 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 Longman (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), 713-739. 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), 103-153. 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), 308-314. doi: 10.1016/j.jfa.2010.03.020.  Google Scholar

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