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Some Results Concerning Hardy and Hardy Type Inequalities

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Handbook of Functional Equations

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 95))

Abstract

We review some recent results concerning functional aspects of the Hardy and Hardy type inequalities. Our main focus is the formulation of such inequalities, for functions having bad behavior at the singularity points. It turns out that Hardy’s singularity terms appear in certain cases as a loss to the Hardy’s functional, while in other cases are additive to it. Surprisingly, in the latter case, Hardy’s functional may be negative. Thus, the validity of the Hardy’s inequality is actually based on these singularity terms.

We also discuss the two topics: nonexistence of \(H_0^1\) minimizers and improved Hardy–Sobolev inequalities. These topics may be seen as a consequence of the connection of the Hardy and Hardy type inequalities with the Sobolev inequality defined in the whole space.

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References

  1. Adams, R.A.: Sobolev Spaces. Academic, New York (1975)

    MATH  Google Scholar 

  2. Adimurthi: Best constants and Pohozaev identity for Hardy–Sobolev-type operators. Commun. Contemp. Math. 15, Article 1250050, 23 p. (2013)

    Google Scholar 

  3. Adimurthi, Esteban, M.J.: An improved Hardy-Sobolev inequality in \(W1,p\) and its application to Schrödinger operators. Nonlinear Differ. Equ. Appl. 12, 243–263 (2005)

    Google Scholar 

  4. Adimurthi, Chaudhuri, N., Ramaswamy, M.: An improved Hardy—Sobolev inequality and its application. Proc. Am. Math. Soc. 130, 489–505 (2002)

    Google Scholar 

  5. Adimurthi, Grossi, M., Santra, S.: Optimal Hardy–Rellich inequalities, maximum principle and related eigenvalue problem. J. Funct. Anal. 240, 36–83 (2006)

    Google Scholar 

  6. Adimurthi, Filippas, S., Tertikas, A.: On the best constant of Hardy–Sobolev inequalities. Nonlinear Anal. 70, 2826–2833 (2009)

    Google Scholar 

  7. Alvinoa, A., Volpicellia, R., Ferone, A.: Sharp Hardy inequalities in the half space with trace remainder term. Nonlinear Anal. 75, 5466–5472 (2012)

    Article  MathSciNet  Google Scholar 

  8. Baras, P., Goldstein, J.A.: The heat equation with a singular potential. Trans. Am. Math. Soc. 284(1) 121–139 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  9. Barbatis, G.: Best constants for higher-order Rellich inequalities in L p. Math Z. 255, 877–896 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Barbatis, G., Tertikas, A.: On a class of Rellich inequalities. J. Comput. Appl. Math. 194, 156–172 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Barbatis, G., Filippas, S., Tertikas, A.: Series expansion for L pHardy inequalities. Indiana Univ. Math. J. 52, 171–190 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Barbatis, G., Filippas, S., Tertikas, A.: Refined geometric L pHardy inequalities. Commun. Contemp. Math. 5, 869–881 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Barbatis, G., Filippas, S., Tertikas, A.: Critical heat kernel estimates for Schrödinger operators via Hardy-Sobolev inequalities. J. Funct. Anal. 208, 1–30 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Berchio, E., Cassani, D., Gazzola, F.: Hardy-Rellich inequalities with boundary remainder terms and applications. Manuscripta Math. 131, 427–458 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Brezis, H., Marcus, M.: Hardy’s inequality revisited. Ann. Sc. Norm. Super. Pisa 25, 217–237 (1997)

    MATH  MathSciNet  Google Scholar 

  16. Brezis, H., Vázquez, J.L.: Blowup solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complutense Madr. 10, 443–469 (1997)

    MATH  Google Scholar 

  17. Burq, N., Planchon, F., Stalker, J.G., Tahvildar-Zadehd, A.S.: Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential. J. Funct. Anal. 203, 519–549 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Cabré, X., Martel, Y.: Existence versus explosion instantané pour des equations de lachaleur linéaires avec potentiel singulier. C. R. Acad. Sci. Paris 329, 973–978 (1999)

    Article  MATH  Google Scholar 

  19. Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53(3), 259–275 (1984)

    MATH  MathSciNet  Google Scholar 

  20. Castorina, D., Fabbri, I., Mancini, G., Sandeep, K.: Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations. J. Differ. Equ. 246, 1187–1206 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Catrina, F., Wang, Z.-Q.: On the Caffarelli–Kohn–Nirenberg inequalities: Sharp constants, existence (and nonexistence) and symmetry of extremal functions. Comm. Pure Appl. Math. LIV, 229–258 (2001)

    Article  MathSciNet  Google Scholar 

  22. Cazacu, C.: Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results. J. Funct. Anal. 263, 3741–3783 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. Chou, K.S., Chu, C.W.: On the best constant for a weighted Sobolev–Hardy inequality. J. Lond. Math. Soc. s2-48(1), 137–151 (1993)

    Article  MathSciNet  Google Scholar 

  24. Cianchi, A., Ferone, A.: Hardy inequalities with non-standard remainder terms. Ann. Inst. Henri Poincaré C Nonlinear Anal 25(5), 889–906 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  26. Davies, E.B.: A review of Hardy inequalities. Oper. Theory Adv. Appl. 110, 55–67 (1999)

    Google Scholar 

  27. Dávila, J., Dupaigne, L.: Hardy-type inequalities. J. Eur. Math. Soc. 6(3), 335–365 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  28. del Pino, M., Dolbeault, J., Filippas, S., Tertikas, A.: A logarithmic Hardy inequality. J. Funct. Anal. 259, 2045–2072 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  29. Ekholm, T., Frank, R.L.: On Lieb–Thirring inequalities for Schrödinger operators with virtual level. Commun. Math. Phys. 264, 725–740 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. Filippas, S., Tertikas, A.: Optimizing improved Hardy inequalities. J. Funct. Anal. 192, 186–233 (2002); Corrigendum. J. Funct. Anal. 255, 2095 (2008)

    Google Scholar 

  31. Filippas, S., Maz’ja, V.G., Tertikas, A.: Critical Hardy–Sobolev inequalities. J. Math. Pures Appl. 87, 37–56 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Frank, R.L., Lieb, E.H., Seiringer, R.: Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators. J. Am. Math. Soc. 21(4), 925–950 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  33. Frank, R.L., Lieb, E.H., Seiringer, R.: Equivalence of Sobolev inequalities and Lieb–Thirring inequalities (2009) (arxiv:0909.5449)

    Google Scholar 

  34. Ghoussoub, N., Moradifam, A.: Bessel potentials and optimal Hardy and Hardy–Rellich inequalities. Math. Ann. 349, 1–57 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  35. Gkikas, K.T.: HardySobolev inequalities in unbounded domains and heat kernel estimates. J. Funct. Anal. 264, 837–893 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  36. Goldstein, J.A., Kombe, I.: The Hardy inequality and nonlinear parabolic equations on Carnot groups. Nonlinear Anal. 69, 4643–4653 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  37. Goldstein, J.A., Zang, Q.S.: Linear parabolic equations with strong singular potentials. Trans. Am. Math. Soc. 355, 197–211 (2003)

    Article  MATH  Google Scholar 

  38. Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1952)

    MATH  Google Scholar 

  39. Karachalios, N.I.: Weyl’s type estimates on the eigenvalues of critical Schrödinger operators. Lett. Math. Phys. 83, 189–199 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  40. Karachalios, N.I., Zographopoulos, N.B.: The semiflow of a reaction diffusion equation with a singular potential. Manuscripta Math. 130, 63–91 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  41. Kombe, I.: Sharp weighted Rellich and uncertainty principle inequalities on Carnot groups. Commun. Appl. Anal. 14, 251–272 (2010)

    MATH  MathSciNet  Google Scholar 

  42. Kovarik, H., Laptev, A.: Hardy inequalities for Robin Laplacians. J. Funct. Anal. 262, 4972–4985 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  43. Krejcirík, D., Zuazua, E.: The Hardy inequality and the heat equation in twisted tubes. J. Math. Pures Appl. 94, 277–303 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  44. Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy Inequality. About Its History and Some Related Results. Vydavatelský Servis, Plzeň (2007)

    MATH  Google Scholar 

  45. Maz’ja, V.G.: Sobolev Spaces. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  46. Moradifam, A.: Optimal weighted Hardy–Rellich inequalities on \(H^2 \cap H^1_0\). J. Lond. Math. Soc. 85(2), 22–40 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  47. Musina, R.: A note on the paper “Optimizing improved Hardy inequalities” by S. Filippas and T. Tertikas [30]. J. Funct. Anal. 256, 2741–2745 (2009)

    Google Scholar 

  48. Nenciu, G., Nenciu, I.: On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in \(\mathbb Rn\). Ann. Henri Poincaré 10, 377–394 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  49. Opic, B., Kufner, A.: Hardy Type Inequalities. Pitman Research Notes in Mathematics, vol. 219. Longman, Harlow (1990)

    MATH  Google Scholar 

  50. Rakotoson, J.-M.: New Hardy inequalities and behaviour of linear elliptic equations. J. Funct. Anal. 263, 2893–2920 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  51. Tertikas, A., Zographopoulos, N.B.: Best constants in the Hardy–Rellich Inequalities and related improvements. Adv. Math. 209, 407–459 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  52. Vancostenoble, J., Zuazua, E.: Null controllability for the heat equation with singular inverse-square potentials. J. Funct. Anal. 254, 1864–1902 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  53. Vázquez, J.L., Zographopoulos, N.B.: Functional aspects of the Hardy inequality. Appearance of a hidden energy. J. Evol. Equ. 12, 713–739 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  54. Vázquez, J.L., Zographopoulos, N.B.: Hardy type inequalities and hidden energies. Discret. Contin. Dyn. Syst. 33(11&12), 5457–5491 (2013)

    Article  MATH  Google Scholar 

  55. Vázquez, J.L., Zuazua, E.: The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. J. Funct. Anal. 173, 103–153 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  56. Zographopoulos, N.B.: Weyl’s type estimates on the eigenvalues of critical Schrödinger operators using improved Hardy–Sobolev inequalities. J. Phys. A Math. Theor. 42, Article 465204 (2009)

    Google Scholar 

  57. Zographopoulos, N.B.: Existence of extremal functions for a Hardy–Sobolev inequality. J. Funct. Anal. 259, 308–314 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Nikolaos B. Zographopoulos

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Correspondence to Nikolaos B. Zographopoulos .

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  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Zographopoulos, N. (2014). Some Results Concerning Hardy and Hardy Type Inequalities. In: Rassias, T. (eds) Handbook of Functional Equations. Springer Optimization and Its Applications, vol 95. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1246-9_20

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