-
Previous Article
Semi linear parabolic equations with nonlinear general Wentzell boundary conditions
- DCDS Home
- This Issue
-
Next Article
Estimating eigenvalues of an anisotropic thermal tensor from transient thermal probe measurements
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 |
  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.
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. |
[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. |
[3] |
H. Brezis and M. Marcus, Hardy's inequality revisited, Ann. Sc. Norm. Pisa, 25 (1997), 217-237. |
[4] |
H. Brezis and J. L. Vázquez, Blowup solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense Madrid, 10 (1997), 443-469. |
[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. |
[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. |
[7] |
E. B. Davies, A review of Hardy inequalities, Oper. Theory Adv. Appl., 110 (1999), 55-67. |
[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. |
[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. |
[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. |
[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. |
[12] |
A. Kufner, L. Maligranda and L.-E. Persson, The Hardy inequality. About its history and some related results, Vydavatelsky' Servis, Plzen, 2007. |
[13] | |
[14] |
R. Musina, A note on the paper [9], J. Funct. Anal., 256 (2009), 2741-2745.
doi: 10.1016/j.jfa.2008.08.009. |
[15] |
B. Opic and A. Kufner, Hardy type inequalities, Pitman Rechearch Notes in Math., 219 Longman (1990). |
[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. |
[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. |
[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. |
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. |
[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. |
[3] |
H. Brezis and M. Marcus, Hardy's inequality revisited, Ann. Sc. Norm. Pisa, 25 (1997), 217-237. |
[4] |
H. Brezis and J. L. Vázquez, Blowup solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense Madrid, 10 (1997), 443-469. |
[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. |
[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. |
[7] |
E. B. Davies, A review of Hardy inequalities, Oper. Theory Adv. Appl., 110 (1999), 55-67. |
[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. |
[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. |
[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. |
[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. |
[12] |
A. Kufner, L. Maligranda and L.-E. Persson, The Hardy inequality. About its history and some related results, Vydavatelsky' Servis, Plzen, 2007. |
[13] | |
[14] |
R. Musina, A note on the paper [9], J. Funct. Anal., 256 (2009), 2741-2745.
doi: 10.1016/j.jfa.2008.08.009. |
[15] |
B. Opic and A. Kufner, Hardy type inequalities, Pitman Rechearch Notes in Math., 219 Longman (1990). |
[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. |
[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. |
[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. |
[1] |
Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, Jiqiang Zheng. The energy-critical NLS with inverse-square potential. Discrete and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387 |
[7] |
Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure and Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258 |
[8] |
Tian Ma, Shouhong Wang. Gravitational Field Equations and Theory of Dark Matter and Dark Energy. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 335-366. doi: 10.3934/dcds.2014.34.335 |
[9] |
Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control and Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011 |
[10] |
Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 597-608. doi: 10.3934/dcds.2020024 |
[11] |
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 and Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073 |
[12] |
Veronica Felli, Ana Primo. Classification of local asymptotics for solutions to heat equations with inverse-square potentials. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 65-107. doi: 10.3934/dcds.2011.31.65 |
[13] |
Barbara Brandolini, Francesco Chiacchio, Cristina Trombetti. Hardy type inequalities and Gaussian measure. Communications on Pure and Applied Analysis, 2007, 6 (2) : 411-428. doi: 10.3934/cpaa.2007.6.411 |
[14] |
Paola Loreti, Daniela Sforza. Inverse observability inequalities for integrodifferential equations in square domains. Evolution Equations and Control Theory, 2018, 7 (1) : 61-77. doi: 10.3934/eect.2018004 |
[15] |
Angelo Alvino, Roberta Volpicelli, Bruno Volzone. A remark on Hardy type inequalities with remainder terms. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 801-807. doi: 10.3934/dcdss.2011.4.801 |
[16] |
Eleftherios Gkioulekas, Ka Kit Tung. Is the subdominant part of the energy spectrum due to downscale energy cascade hidden in quasi-geostrophic turbulence?. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 293-314. doi: 10.3934/dcdsb.2007.7.293 |
[17] |
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 |
[18] |
Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649 |
[19] |
Toshiyuki Suzuki. Semilinear Schrödinger evolution equations with inverse-square and harmonic potentials via pseudo-conformal symmetry. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4347-4377. doi: 10.3934/cpaa.2021163 |
[20] |
Annalisa Cesaroni, Matteo Novaga. Volume constrained minimizers of the fractional perimeter with a potential energy. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 715-727. doi: 10.3934/dcdss.2017036 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]