Role of the fundamental solution in Hardy-Sobolev-type inequalities

Adimurthi, ; Sekar, Anusha (2006) Role of the fundamental solution in Hardy-Sobolev-type inequalities Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 136 (6). pp. 1111-1130. ISSN 0308-2105

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Official URL: http://journals.cambridge.org/action/displayAbstra...

Related URL: http://dx.doi.org/10.1017/S030821050000490X

Abstract

Let n ≥ 3, Ω⊂ Rn be a domain with 0∈ Ω, then, for all u∈ H10 Ω, the Hardy-Sobolev inequality says that ∫Ω|∇u|2-(n-2/2)2Ωu2/|x|2≥0 and equality holds if and only if u = 0 and ((n-2)/2)2 is the best constant which is never achieved. In view of this, there is scope for improving this inequality further. In this paper we have investigated this problem by using the fundamental solutions and have obtained the optimal estimates. Furthermore, we have shown that this technique is used to obtain the Hardy-Sobolev type inequalities on manifolds and also on the Heisenberg group.

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ID Code:11858
Deposited On:13 Nov 2010 13:44
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