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 | References | Similar Articles | Additional Information

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 $\mu _p(\Omega )=\inf _{W_{1, p}(\Omega )}\left (\int _\Omega |\nabla u|^p / \int _\Omega |u/\delta |^p \right )$ 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.

- Shmuel Agmon,
*On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds*, Methods of functional analysis and theory of elliptic equations (Naples, 1982) Liguori, Naples, 1983, pp. 19–52. MR**819005** - Shmuel Agmon,
*Bounds on exponential decay of eigenfunctions of Schrödinger operators*, Schrödinger operators (Como, 1984) Lecture Notes in Math., vol. 1159, Springer, Berlin, 1985, pp. 1–38. MR**824986**, DOI https://doi.org/10.1007/BFb0080331 - Shmuel Agmon,
*A representation theorem for solutions of Schrödinger type equations on noncompact Riemannian manifolds*, Astérisque**210**(1992), 5, 13–26. Méthodes semi-classiques, Vol. 2 (Nantes, 1991). MR**1221349** - S. Agmon, Personal communication.
- Alano Ancona,
*On strong barriers and an inequality of Hardy for domains in ${\bf R}^n$*, J. London Math. Soc. (2)**34**(1986), no. 2, 274–290. MR**856511**, DOI https://doi.org/10.1112/jlms/s2-34.2.274 - Giuseppe Buttazzo and Victor J. Mizel,
*On a gap phenomenon for isoperimetrically constrained variational problems*, J. Convex Anal.**2**(1995), no. 1-2, 87–101. MR**1363362** - D. K. Ganguly and D. Bandyopadhyay,
*Approximation of fixed points in Banach space by iteration processes using infinite matrices*, Soochow J. Math.**22**(1996), no. 3, 395–403. MR**1406511** - E. B. Davies,
*The Hardy constant*, Quart. J. Math. Oxford Ser. (2)**46**(1995), no. 184, 417–431. MR**1366614**, DOI https://doi.org/10.1093/qmath/46.4.417 - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190** - G. H. Hardy, Note on a Theorem of Hilbert,
*Math. Zeit.***6**(1920), 314–317. - G. H. Hardy, An inequality between integrals,
*Messenger of Math.***54**(1925), 150–156. - Lars Hörmander,
*Notions of convexity*, Progress in Mathematics, vol. 127, Birkhäuser Boston, Inc., Boston, MA, 1994. MR**1301332** - E. Landau, A note on a theorem concerning series of positive terms,
*J. London Math. Soc.,***1**(1926), 38–39. - I. F. Lezhenina and P. E. Sobolevskiĭ,
*Elliptic and parabolic boundary value problems with a singular coefficient estimate*, Dokl. Akad. Nauk Ukrain. SSR Ser. A**3**(1989), 27–31, 88 (Russian, with English summary). MR**1001060** - Tanya Matskewich and Pavel E. Sobolevskii,
*The best possible constant in generalized Hardy’s inequality for convex domain in ${\bf R}^n$*, Nonlinear Anal.**28**(1997), no. 9, 1601–1610. MR**1431208**, DOI https://doi.org/10.1016/S0362-546X%2896%2900004-1 - B. Opic and A. Kufner,
*Hardy-type inequalities*, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990. MR**1069756** - J. Serrin,
*The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables*, Philos. Trans. Roy. Soc. London Ser. A**264**(1969), 413–496. MR**282058**, DOI https://doi.org/10.1098/rsta.1969.0033 - Andreas Wannebo,
*Hardy inequalities*, Proc. Amer. Math. Soc.**109**(1990), no. 1, 85–95. MR**1010807**, DOI https://doi.org/10.1090/S0002-9939-1990-1010807-1

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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

MR Author ID:
139695

Email:
pincho@tx.technion.ac.il

Keywords:
Rayleigh quotient,
concentration effect,
essential spectrum.

Received by editor(s):
September 5, 1996

Article copyright:
© Copyright 1998
American Mathematical Society