Sharp Sobolev inequalities of second order |
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Authors: | Email author" target="_blank">Emmanuel?HebeyEmail author |
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Institution: | (1) Département de Mathématiques, Université de Cergy-Pointoise, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France |
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Abstract: | Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and
2
2
(M) be the Sobolev space consisting of functions in L2(M) whose derivatives up to the order two are also in L2(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H
2
2
(M), where 2#=2n/(n−4) is critical, and
is the usual norm on the Sobolev space H
1
2
(M) consisting of functions in L2(M) whose derivatives of order one are also in L2(M). The sharp constant A in this inequality is K
0
2
where K0, an explicit constant depending only on n, is the sharp constant for the Euclidean Sobolev inequality
. We prove in this article that for any compact Riemannian manifold, A=K
0
2
is attained in the above inequality. |
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Keywords: | Math Subject Classifications" target="_blank">Math Subject Classifications 58E35 |
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