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Sharp constant in Nash's inequality

yhhuang 添加于 2011-9-29 04:47 | 2398 次阅读 | 0 个评论

作者
Carlen, Eric A., Loss, Michael
摘要
From the text: "The Nash inequality states that (1) $$\Big(\int_{\bold R^n}|f(x)|^2\,d^nx\Big)^{1+2/n}\leq C_n\int_{\bold R^n}|\nabla f(x)|^2\,d^nx\Big(\int_{\bold R^n}|f(x)|\,d^nx\Big)^{4/n}$$ (∫Rn|f(x)|2dnx)1+2/n≤Cn∫Rn|∇f(x)|2dnx(∫Rn|f(x)|dnx)4/n for a constant $C_n$Cn depending only on $n$n. This inequality is a particular case of the Gagliardo-Nirenberg inequalities for which numerous applications have been found. In this note we compute the sharp constant in (1) and determine all of the cases of equality. A particularly striking feature of the result is that all of the extremals have compact support.''
详细资料
- 文献种类: Journal Article
- 期刊名称: International Mathematics Research Notices
- 期刊缩写: Internat. Math. Res. Notices
- 期卷页: 1993年第7卷 213-215页
学科领域自然科学 » 数学
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inequalities
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Sharp constant in Nash's inequality

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The interpolation between different norms:

$\|u\|_2, \|u\|_1, \|\nabla u\|_2$ .

With the assumption of the minimizer is obtained for radially decreasing function, choose a ball, whose radius is given by self-consistence conditions.

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Sharp constant in Nash's inequality

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