Ngô Quốc Anh

June 7, 2010

Some important functional inequalities, 2

This entry can be considered as a continued part to a recent entry where lots of significantly important inequalities (Hardy, Opial, Rellich, Serrin, Caffarelli–Kohn–Nirenberg, Gagliardo-Nirenberg-Sobolev, Horgan) have been considered.

Today we shall continue to list here more important inequalities in the literature. Given 1\leqq q < n, 0 < \theta \leqq 1, s>1, define the number r by

\displaystyle\frac{1}{s} - \frac{1}{r} = \theta \left( {\frac{1}{s} - \frac{1}{{{q^ \star }}}} \right)

where

\displaystyle q^\star=\frac{n-q}{nq}.

Gagliardo-Nirenberg’s inequality. For any u \in C_0^\infty(\mathbb R^n) one has

\displaystyle {\left( {\int_{{\mathbb{R}^n}} {|u{|^r}dx} } \right)^{\frac{1}{r}}} \leqq C{\left( {\int_{{\mathbb{R}^n}} {|\nabla u{|^q}dx} } \right)^{\frac{\theta }{q}}}{\left( {\int_{{\mathbb{R}^n}} {|u{|^s}dx} } \right)^{\frac{{1 - \theta }}{s}}}.

When \theta=1 and r=q^\star, Gagliardo-Nirenberg’s inequality then becomes the well-known Sobolev inequality.

Sobolev’s inequality. For any u \in C_0^\infty(\mathbb R^n) one has

\displaystyle {\left( {\int_{{\mathbb{R}^n}} {|u{|^{{q^ \star }}}dx} } \right)^{\frac{1}{{{q^ \star }}}}} \leqq K(n,q){\left( {\int_{{\mathbb{R}^n}} {|\nabla u{|^q}dx} } \right)^{\frac{1}{q}}}.

The best constant K(n,q) has been obtained by Aubin [here] and Talenti [here], independently. Namely, they showed that

\displaystyle K(n,1) = \frac{1}{n}\omega _n^{ - \frac{1}{n}}

and

\displaystyle K(n,q) = \frac{1}{n}{\left( {\frac{{n(q - 1)}}{{n - q}}} \right)^{\frac{{q - 1}}{q}}}{\left( {\frac{{\Gamma (n + 1)}}{{n{\omega _n}\Gamma \left( {\frac{n}{q}} \right)\Gamma \left( {n + 1 - \frac{n}{q}} \right)}}} \right)^{\frac{1}{n}}}, \quad q > 1

where \omega_n is the volume of the unit ball in \mathbb R^n and \Gamma the gamma function.

When \theta=\frac{n}{n+2}, q=2, and r=2, Gagliardo-Nirenberg’s inequality then becomes the well-known Nash inequality.

Nash’s inequality. For any u \in C_0^\infty(\mathbb R^n) one has

\displaystyle {\left( {\int_{{\mathbb{R}^n}} {|u{|^2}dx} } \right)^{1 + \frac{2}{n}}} \leqq C(n)\left( {\int_{{\mathbb{R}^n}} {|\nabla u{|^2}dx} } \right){\left( {\int_{{\mathbb{R}^n}} {|u|dx} } \right)^{\frac{4}{n}}}.

The best constant C(n) for the Nash inequality is given by

\displaystyle C(n) = \frac{{2{{\left( {\frac{{n + 2}}{2}} \right)}^{\frac{{n + 2}}{n}}}}}{{n\lambda _1^N\omega _n^{\frac{2}{n}}}}

where \lambda _1^N is the first non-zero Neumann eigenvalue of the Laplacian operator in the unit ball.  This come from a joint work between Carlen and Loss [here].

Another consequence of Gagliardo-Nirenberg’s inequality is the logarithmic Sobolev inequality.

logarithmic Sobolev’s inequality. For any u \in C_0^\infty(\mathbb R^n) one has

\displaystyle\int_{{\mathbb{R}^n}} {{u^2}{{(\log u)}^2}dx} \leqslant \frac{n}{2}\log \left( {\widetilde C\int_{{\mathbb{R}^n}} {|\nabla u{|^2}dx} } \right)

where u also satisfies

\displaystyle\int_{{\mathbb{R}^n}} {{u^2}dx} = 1.

In fact, it can be obtained as the limit case when \theta \to 0, that is, r=2 and s \to r, s<r. To see this, let us first notice the fact that the constant C in Gagliardo-Nirenberg’s inequality is independent of s. We can rewrite Gagliardo-Nirenberg’s inequality as

\displaystyle{\left( {\frac{{{{\left\| u \right\|}_{{L^r}}}}}{{{{\left\| u \right\|}_{{L^s}}}}}} \right)^{\frac{1}{{\frac{1}{s} - \frac{1}{r}}}}} \leqslant {\left( {\frac{{{C_0}{{\left\| {\nabla u} \right\|}_{{L^q}}}}}{{{{\left\| u \right\|}_{{L^s}}}}}} \right)^{\frac{1}{{\frac{1}{s} - \frac{1}{{{q^ \star }}}}}}}

where C_0=C^\frac{1}{\theta}. It then follows that

\displaystyle\frac{{\log {{\left\| u \right\|}_{{L^r}}} - {{\left\| u \right\|}_{{L^s}}}}}{{\frac{1}{s} - \frac{1}{r}}} \leqslant \frac{1}{{\frac{1}{s} - \frac{1}{{{q^ \star }}}}}\log \left( {{C_0}\frac{{{{\left\| {\nabla u} \right\|}_{{L^q}}}}}{{{{\left\| u \right\|}_{{L^s}}}}}} \right).

Thus when s \to r^- we get

\displaystyle\int_{{\mathbb{R}^n}} {\left[ {{u^r}\log {{\left( {\frac{u}{{{{\left\| u \right\|}_{{L^r}}}}}} \right)}^r}} \right]dx} \leqq \frac{1}{{\frac{1}{s} - \frac{1}{{{q^ \star }}}}}\left\| u \right\|_{{L^r}}^r\log \left( {{C_0}\frac{{{{\left\| {\nabla u} \right\|}_{{L^q}}}}}{{{{\left\| u \right\|}_{{L^r}}}}}} \right)

where we have used the fact that the function

\displaystyle\varphi (u) = \log {\left\| u \right\|_{{L^u}}}

satisfies

\displaystyle - \left\| u \right\|_{{L^r}}^r\varphi '\left( {\frac{1}{r}} \right) = \int_{{\mathbb{R}^n}} {\left[ {{u^r}\log {{\left( {\frac{u}{{{{\left\| u \right\|}_{{L^r}}}}}} \right)}^r}} \right]dx} .

Therefore, replacing r = q = 2 and writing \widetilde C = \sqrt{C_0}, we obtain the logarithmic Sobolev’s inequality.

The best constant for the logarithmic Sobolev inequality is given by

\displaystyle\widetilde C(n) = \frac{2}{{n\pi e}}.

We refer the reader to a book due to Hebey entitled “Nonlinear analysis on manifolds: Sobolev spaces and inequalities” for details.

The best constant for the Gagliardo-Nirenberg inequality is not completely solved. In some cases, we was able to find its best constants [here, here].

See also: Sobolev type inequalities on Riemannian manifolds

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