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), 32373255
MSC (1991):
Primary 49R05, 35J70
MathSciNet review:
1458330
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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 onedimensional 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:
http://dx.doi.org/10.1090/S0002994798021229
PII:
S 00029947(98)021229
Keywords:
Rayleigh quotient,
concentration effect,
essential spectrum.
Received by editor(s):
September 5, 1996
Article copyright:
© Copyright 1998
American Mathematical Society
