Proceedings of the Japan Academy, Series A, Mathematical Sciences

Missing terms in Hardy-Sobolev inequalities

Hiroshi Ando, Alnar Detalla, and Toshio Horiuchi

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Abstract

In this article we shall investigate Hardy-Sobolev inequalities and improve them by adding a term with a singular weight of the type $(\log(1/|x|))^{-2}$. We show that this weight function is optimal in the sense that the inequality fails for any other weight more singular than this one. As an application, we use our improved inequality to study the weighted eigenvalue problem stated in §5.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 80, Number 8 (2004), 160-165.

Dates
First available in Project Euclid: 18 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1116442375

Digital Object Identifier
doi:10.3792/pjaa.80.160

Mathematical Reviews number (MathSciNet)
MR2099744

Zentralblatt MATH identifier
1112.35066

Subjects
Primary: 35J70: Degenerate elliptic equations
Secondary: 35J60: Nonlinear elliptic equations

Keywords
Hardy-Sobolev inequality eigenvalue

Citation

Detalla, Alnar; Horiuchi, Toshio; Ando, Hiroshi. Missing terms in Hardy-Sobolev inequalities. Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 8, 160--165. doi:10.3792/pjaa.80.160. https://projecteuclid.org/euclid.pja/1116442375


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References

  • Adimurthi, Chaudhuri, N., and Ramaswamy, M.: An improved Hardy-Sobolev inequality and its application. Proc. Amer. Math. Soc., 130(2), 489–505 (2002).
  • Detalla, A., Horiuchi, T., and Ando, H.: Missing terms in Hardy-Sobolev inequalities and its application. Far East J. Math. Sci., 14(3), 333-359 (2004).
  • Boccardo, L., and Murat, F.: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19(6), 581–597 (1992).
  • Horiuchi, T.: Missing terms in Generalized Hardy's inequalities and its applications. J. Math. Kyoto Univ., 43(2), 235–260 (2003).
  • Struwe, M.: Variational Methods. Applied to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer, Berlin (1996).
  • Talenti, G.: Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces. Ann. Mat. Pura Appl. (4), 120, 160–184 (1979).