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Nash’s inequality

yhhuang 添加于 2012-6-3 11:22 | 1914 次阅读 | 0 个评论

Several related inequalities with sharp constants

Sobolev inequalities of the type (completely solved)
$\|u\|_{L^{p^*}} \leq K \|\nabla u\|_{p}$
Gagliardo-Nirenberg
$\|u\|_r \leq K_n(q,r,s)\|\nabla u\|_q^\theta\|u\|_s^{1-\theta}$
The connection through nonlinear Schrodinger equation: for example the case q=2, s=2 is related to the solution of
$\frac{n\sigma}{2}\Delta \psi-\frac{2+2\sigma-n\sigma}{2}\psi+\psi^{2\sigma+1}=0$
The case when p in the sobolev inequality is given by the Moser-Trudinger inequality: if
$\int_D |\nabla f(x)|^ndx \leq 1$
then
$\frac{1}{|D|}\int_D e^{\alpha |f(x)|^{n/(n-1)}}dx \leq c_n$ .