On the best constant for Hardy's inequality in

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 be a domain in and . We consider the (generalized) Hardy inequality , where . 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 and the *existence* of a minimizer for this Rayleigh quotient. It is shown that for all smooth -dimensional domains, , where is the one-dimensional Hardy constant. Moreover it is shown that for all those domains *not* possessing a minimizer for the above Rayleigh quotient. Finally, for , it is proved that if and only if the Rayleigh quotient possesses a minimizer. Examples show that strict inequality may occur even for bounded smooth domains, but for convex domains.

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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