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Optimizing Improved Hardy Inequalities

✍ Scribed by Stathis Filippas; Achilles Tertikas

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 354 KB
Volume: 192
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.2001.3900

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✦ Synopsis

Let O be a bounded domain in R N ; N 53, containing the origin. Motivated by a question of Brezis and V! a azquez, we consider an Improved Hardy Inequality with best constant b, that we formally write as:

We first give necessary conditions on the potential V , under which the previous inequality can or cannot be further improved. We show that the best constant b is never achieved in H 1 0 ðOÞ, and in particular that the existence or not of further correction terms is not connected to the nonachievement of b in H 1 0 ðOÞ. Our analysis reveals that the original inequality can be repeatedly improved by adding on the right-hand side specific potentials. This leads to an infinite series expansion of Hardy's inequality. The series obtained is in some sense optimal. In establishing these results we derive various sharp improved Hardy-Sobolev Inequalities.

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