Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality: $(∫|u|^{r}dx)^{\frac{1}{r}} \leq GN(N,p,q,r)(∫|∇u|^{p}dx)^{\frac{a}{p}}(∫|u|^{q}dx)^{\frac{1-a}{q}}$. Can the best constant GN(N,p,q,r) be achieved in some cases? Does anyone know good references about this subject?
$\begingroup$
$\endgroup$
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
2
There is a $1$--parameter family of inequalities where the sharp constants and corresponding extremal functions are known. I believe this was first established by Del Pino and Dolbeault. Cordero, Nazaret, and Villani gave a beautiful optimal transportation proof. See their paper for the relevant references. The family includes the sharp Sobolev and sharp log-Sobolev inequalities.
ADDED: Here are the exactly references:
Del Pino, Manuel(RCH-UCSP-EM); Dolbeault, Jean(F-PARIS9-A) Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. (9) 81 (2002), no. 9, 847–875.
Cordero-Erausquin, D.(F-MARN-AMA); Nazaret, B.(F-ENSLY-PM); Villani, C.(F-ENSLY-PM) A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182 (2004), no. 2, 307–332.
answered Jun 27, 2015 at 4:12
-
$\begingroup$ Thanks very much, Deane. In the papers you gave, the authors can calculate the best constant GN(N,p,q,r) and extremal functions explicitly, of course for 1--parameter family of inequalities. So just assume that we don't care what the value of GN(N,p,q,r) is, but just the question that if GN(N,p,q,r) can be attained, can we show it in general? $\endgroup$ Commented Jun 28, 2015 at 2:46
-
$\begingroup$ For $\mathbb{R}^n$ or any other noncompact Riemannian manifold, I'm not sure. It's basically a question of showing that an appropriately normalized minimizing sequence of functions is compact with respect to some topology and that the limit is nonzero. The main issue here is making sure that the "mass" of the function does not leak out to infinity. If the domain is a closed Riemannian manifold, then the answer is yes. $\endgroup$ Commented Jun 28, 2015 at 18:44
$\begingroup$
$\endgroup$
2
As a complement to Deane Yang's answer: arguably a special case of major interest is $p = q = 2$. For this case, existence of Gagliardo-Nirenberg maximizers was proved in $H^1(\mathbb{R}^n)$ by Michael Weinstein in this paper: http://www.ams.org/mathscinet-getitem?mr=691044. The basic argument was, using spatial scaling in $\mathbb{R}^n$, consider a maximizing sequence $\varphi_n$ such that $\Vert \varphi_n\Vert_{L^2} = 1 = \Vert \nabla \varphi_n\Vert_{L^2} $. Then, since $\varphi_n$ is weakly $H^1$-bounded, the weak limit $u$ is shown to be (using Sobolev embedding) the maximizer. Note that this approach does not work for the hyperbolic space, for example, where spatial dilation is not available. The preprint http://arxiv.org/abs/1406.4931 shows that the GN maximizer ($p = q = 2$) does not exist in $H^1(\mathbb{H}^n)$.
-
$\begingroup$ Thanks very much. Do you know any references showing the existence of Gagliardo-Nirenberg maximizers in other cases? Of course, finding what GN(N,p,q,r) is is difficult, but I think that showing GN(N,p,q,r) can be attained is much easier, right? $\endgroup$ Commented Jun 28, 2015 at 2:52
-
$\begingroup$ Actually I do not. And actually showing that the best constant of the GN inequality can be attained can be quite tricky because of the global nature of the problem. Perhaps this is not a good analogy, but you can read up on the Yamabe problem to appreciate how difficult it can be to establish the existence of extremizers of nonlinear functionals. $\endgroup$– SMSCommented Jun 28, 2015 at 3:05