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Transactions of the American Mathematical Society

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On the best constant for Hardy's inequality in $\mathbb R^n$


Authors: Moshe Marcus, Victor J. Mizel and Yehuda Pinchover
Journal: Trans. Amer. Math. Soc. 350 (1998), 3237-3255
MSC (1991): Primary 49R05, 35J70
DOI: https://doi.org/10.1090/S0002-9947-98-02122-9
MathSciNet review: 1458330
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Abstract: Let $\Omega $ be a domain in $\mathbb R^n$ and $p\in (1,\infty)$. We consider the (generalized) Hardy inequality $\int _\Omega |\nabla u|^p\geq K\int _\Omega |u/\delta |^p$, where $\delta (x)=\operatorname{dist}\,(x,\partial \Omega )$. The inequality is valid for a large family of domains, including all bounded domains with Lipschitz boundary. We here explore the connection between the value of the Hardy constant $\mu _p(\Omega )=\inf _{\stackrel{\circ}{W}_{1,p}(\Omega )}\left (\int _\Omega |\nabla u|^p\,/\,\int _\Omega |u/\delta |^p \right )$ and the existence of a minimizer for this Rayleigh quotient. It is shown that for all smooth $n$-dimensional domains, $\mu _p(\Omega )\leq c_p$, where $c_p=(1-{1\over p})^p$ is the one-dimensional Hardy constant. Moreover it is shown that $\mu _p(\Omega )=c_p$ for all those domains not possessing a minimizer for the above Rayleigh quotient. Finally, for $p=2$, it is proved that $\mu _2(\Omega )<c_2=1/4$ if and only if the Rayleigh quotient possesses a minimizer. Examples show that strict inequality may occur even for bounded smooth domains, but $\mu _p=c_p$ for convex domains.


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  • 1. S. Agmon, On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, in ``Methods of Functional Analysis and the Theory of Elliptic Equations'', ed. D. Greco, Liguori, Naples, 1982, 19-52. MR 87b:58087
  • 2. S. Agmon, Bounds on exponential decay of eigenfunctions, in ``Schrödinger Operators'', ed. S. Graffi, Lecture Notes in Math., Vol. 1159, Springer-Verlag, Berlin, 1985, pp. 1-38. MR 87i:35157
  • 3. S. Agmon, A representation theorem for solutions of Schrödinger type equations on non-compact Riemannian manifolds, Astérisque, 210 (1992), 13-26. MR 94g:58203
  • 4. S. Agmon, Personal communication.
  • 5. A. Ancona, On strong barriers and an inequality of Hardy for domains in $\mathbb R^n$, J. London Math. Soc. (2) 34 (1986), 274-290. MR 87k:31004
  • 6. G. Buttazzo and V. J. Mizel, On a gap phenomenon for isoperimetrically constrained variational problems, J. Conv. Analysis, 2 (1995), 87-101. MR 96j:49001
  • 7. E. B. Davies, ``Spectral Theory and Differential Operators'', Cambridge Univ. Press, Cambridge, 1995. MR 97h:47056
  • 8. E. B. Davies, The Hardy constant, Quart. J. Math. Oxford (2) 46 (1995), 417-431. MR 97b:46041
  • 9. D. Gilbarg and N. S. Trudinger, ``Elliptic Partial Differential Equations of the Second Order'', 2nd edition, Springer-Verlag, New York, 1983. MR 86c:35035
  • 10. G. H. Hardy, Note on a Theorem of Hilbert, Math. Zeit. 6 (1920), 314-317.
  • 11. G. H. Hardy, An inequality between integrals, Messenger of Math. 54 (1925), 150-156.
  • 12. L. Hörmander, ``Notions of Convexity'', Birkhäuser, Boston, 1994. MR 95k:00002
  • 13. E. Landau, A note on a theorem concerning series of positive terms, J. London Math. Soc., 1 (1926), 38-39.
  • 14. I. F. Lezhenina and P. E. Sobolevskii, Elliptic and parabolic boundary value problems with singular estimate of coefficients, Dokl. Akad. Nauk Ukrain. SSR, Ser A, 1989, no. 3, 27-31 (Russian). MR 90d:35071
  • 15. T. Matskewich and P. E. Sobolevskii, The best possible constant in a generalized Hardy's inequality for convex domains in $\mathbb R^n$, Nonlinear Analysis TMA, 28 (1997), 1601-1610. MR 98a:26019
  • 16. B. Opic and A. Kufner, ``Hardy-type Inequalities'', Pitman Research Notes in Math., Vol. 219, Longman 1990. MR 92b:26028
  • 17. J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London, Ser. A, 264 (1969), 413-469. MR 43:7772
  • 18. A. Wannebo, Hardy inequalities, Proc. Amer. Math. Soc. 109 (1990), 85-95. MR 90h:26025

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Additional Information

Moshe Marcus
Affiliation: Department of Mathematics, Technion, Haifa, Israel
Email: marcusm@tx.technion.ac.il

Victor J. Mizel
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email: vm09+@andrew.cmu.edu

Yehuda Pinchover
Affiliation: Department of Mathematics, Technion, Haifa, Israel
Email: pincho@tx.technion.ac.il

DOI: https://doi.org/10.1090/S0002-9947-98-02122-9
Keywords: Rayleigh quotient, concentration effect, essential spectrum.
Received by editor(s): September 5, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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