Mathematische Annalen

, Volume 349, Issue 1, pp 1–57 | Cite as

Bessel pairs and optimal Hardy and Hardy–Rellich inequalities

Article

Abstract

We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in Rn, n ≥ 1, so that the following inequalities hold for all \({u \in C_{0}^{\infty}(B)}\) :
$$\label{one} \int\limits_{B}V(x)|\nabla u |^{2}dx \geq \int\limits_{B} W(x)u^2dx,$$
$$\label{two} \int\limits_{B}V(x)|\Delta u |^{2}dx \geq\int\limits_{B} W(x)|\nabla u|^{2}dx+(n-1)\int\limits_{B}\left(\frac{V(x)}{|x|^2}-\frac{V_r(|x|)}{|x|}\right)|\nabla u|^2dx.$$
This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behaviour of certain ordinary differential equations, and helps in the identification of a large number of such couples (V, W)—that we call Bessel pairs—as well as the best constants in the corresponding inequalities. This allows us to improve, extend, and unify many results—old and new—about Hardy and Hardy–Rellich type inequalities, such as those obtained by Caffarelli et al. (Compos Math 53:259–275, 1984), Brezis and Vázquez (Revista Mat. Univ. Complutense Madrid 10:443–469, 1997), Wang and Willem (J Funct Anal 203:550–568, 2003), Adimurthi et al. (Proc Am Math Soc 130:489–505, 2002), and many others.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adimurthi, Chaudhuri N., Ramaswamy N.: An improved Hardy Sobolev inequality and its applications. Proc. Am. Math. Soc. 130, 489–505 (2002)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Adimurthi, Grossi M., Santra S.: Optimal Hardy-Rellich inequalities, maximum principles and related eigenvalue problems. J. Funct. Anal. 240, 36–83 (2006)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Agueh M., Ghoussoub N., Kang X.S.: Geometric inequalities via a general comparison principle for interacting gases. Geom. Funct. Anal. 14(1), 215–244 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barbatis G.: Best constants for higher-order Rellich inequalities in L P(Ω). Math. Z. 255, 877–896 (2007)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Beckner W.: Weighted inequalities and Stein-Weiss potentials. Forum Math. 20, 587–606 (2008)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Blanchet A., Bonforte M., Dolbeault J., Grillo G., Vasquez J.L.: Hardy-Poincaré inequalities and applications to nonlinear diffusions. C. R. Acad. Sci. Paris, Ser. I 344, 431–436 (2007)MATHGoogle Scholar
  7. 7.
    Brezis H., Lieb E.H.: Sobolev inequalities with remainder terms. J. Funct. Anal. 62, 73–86 (1985)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Brezis H., Marcus M.: Hardy’s inequality revisited. Ann. Scuola. Norm. Sup. Pisa 25, 217–237 (1997)MATHMathSciNetGoogle Scholar
  9. 9.
    Brezis H., Marcus M., Shafrir I.: Extremal functions for Hardy’s inequality with weight. J. Funct. Anal. 171, 177–191 (2000)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Brezis H., Vázquez J.L.: Blowup solutions of some nonlinear elliptic problems. Revista Mat. Univ. Complutense Madrid 10, 443–469 (1997)MATHGoogle Scholar
  11. 11.
    Caffarelli L., Kohn R., Nirenberg L.: First order interpolation inequalities with weights. Compos. Math. 53, 259–275 (1984)MATHMathSciNetGoogle Scholar
  12. 12.
    Catrina F., Wang Z.-Q.: On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Commun. Pure Appl. Math. 54, 229–258 (2001)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Cordero-Erausquin D., Nazaret B., Villani C.: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182(2), 307–332 (2004)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Cowan, C., Esposito, P., Ghoussoub, N., Moradifam, A.: The critical dimension for a fourth order elliptic problem with singular nonlinearity. Arch. Ration. Mech. Anal. (to appear)Google Scholar
  15. 15.
    Davies E.B.: A review of Hardy inequalities. Oper. Theory Adv. Appl. 110, 55–67 (1999)Google Scholar
  16. 16.
    Davies E.B., Hinz A.M.: Explicit constants for Rellich inequalities in L p(Ω). Math. Z. 227, 511–523 (1998)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Esposito, P., Ghoussoub, N., Guo, Y.J.: Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, 320 pp. Courant Institute Lecture Notes, AMS (2010)Google Scholar
  18. 18.
    Filippas S., Tertikas A.: Optimizing improved Hardy inequalities. J. Funct. Anal. 192(1), 186–233 (2002)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Fleckinger J., Harrell E.M. II, Thelin F.: Boundary behaviour and estimates for solutions of equations containing the p-Laplacian. Electron. J. Differ. Equ. 38, 1–19 (1999)Google Scholar
  20. 20.
    Ghoussoub N., Moradifam A.: On the best possible remaining term in the Hardy inequality. Proc. Nat. Acad. Sci. 105(37), 13746–13751 (2008)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Hartman P.: Ordinary Differential Equations. Wiley, New York (1964)MATHGoogle Scholar
  22. 22.
    Huang C.: Oscillation and Nonoscillation for second order linear differential equations. J. Math. Anal. Appl. 210, 712–723 (1997)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Liskevich V., Lyakhova S., Moroz V.: Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains. J. Differ. Equ. 232, 212–252 (2007)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Moradifam A.: On the critical dimension of a fourth order elliptic problem with negative exponent. J. Differ. Equ. 248, 594–616 (2010)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Moradifam A.: The singular extremal solutions of the bilaplacian with exponential nonlinearity. Proc. Am. Math. Soc. 138, 1287–1293 (2010)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Moradifam, A.: Optimal weighted Hardy-Rellich inequalities on \({H^2\cap H^1_0}\) . (submitted)Google Scholar
  27. 27.
    Opic, B., Kufner, A.: Hardy type inequalities. In: Pitman Research Notes in Mathematics, vol. 219. Longman, New York (1990)Google Scholar
  28. 28.
    Peral I., Vázquez J.L.: On the stability and instability of the semilinear heat equation with exponential reaction term. Arch. Ration. Mech. Anal. 129, 201–224 (1995)MATHCrossRefGoogle Scholar
  29. 29.
    Simon B.: Schrödinger semigroups. Bull. Am. Math. Soc. 7, 447–526 (1982)MATHCrossRefGoogle Scholar
  30. 30.
    Sugie J., Kita K., Yamaoka N.: Oscillation constant of second-order non-linear self-adjoint differential equations. Ann. Mat. Pura Appl. 181(4), 309–337 (2002)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Tertikas A.: Critical phenomena in linear elliptic problems. J. Funct. Anal. 154, 42–66 (1998)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Tertikas A., Zographopoulos N.B.: Best constants in the Hardy-Rellich inequalities and related improvements. Adv. Math. 209, 407–459 (2007)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Vázquez J.L.: Domain of existence and blowup for the exponential reaction diffusion equation. Indiana Univ. Math. J. 48, 677–709 (1999)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    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)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Wang Z.-Q., Willem M.: Caffarelli-Kohn-Nirenberg inequalities with remainder terms. J. Funct. Anal. 203, 550–568 (2003)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Wintner A.: On the nonexistence of conjugate points. Am. J. Math. 73, 368–380 (1951)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Wintner A.: On the comparision theorem of Knese-Hille. Math. Scand. 5, 255–260 (1957)MATHMathSciNetGoogle Scholar
  38. 38.
    Wong J.S.W.: Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients. Trans. Am. Math. Soc. 144, 197–215 (1969)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada

Personalised recommendations