Ngô Quốc Anh

July 8, 2010

The Moser-Trudinger inequality

Followed by an entry where the Trudinger inequality had been discussed we now consider an important variant of it known as the Moser-Trudinger inequality.

Let us remind the Trudinger inequality

Theorem (Trudinger). Let $\Omega \subset \mathbb R^n$ be a bounded domain and $u \in W_0^{1, n}(\Omega)$ with $\displaystyle\int_\Omega {{{\left| {\nabla u} \right|}^n}dx} \leqslant 1$.

Then there exist universal constants $\beta>0$, $C_1>0$ such that $\displaystyle\int_\Omega {\exp (\beta {|u|^\frac{n}{n-1}})dx} \leqslant {C_1}|\Omega |$.

The Trudinger inequality has lots of application. For application to the prescribed Gauss curvature equation, one requires a particular value for the best constant $\beta_0$. In connection with his work on the Gauss curvature equation, J. Moser [here] sharpended the above result of Trungdier as follows

Theorem (Moser). Let $\Omega \subset \mathbb R^n$ be a bounded domain and $u \in W_0^{1, n}(\Omega)$ with $\displaystyle\int_\Omega {{{\left| {\nabla u} \right|}^n}dx} \leqslant 1$.

Then there exist sharp constants $\beta_0=\beta(n)>0$, $C_1=C_1(n)>0$ given by $\displaystyle \beta_0=n\omega_{n-1}^{\frac{1}{n}-1}$

such that $\displaystyle\int_\Omega {\exp (\beta {|u|^\frac{n}{n-1}})dx} \leqslant {C_1}|\Omega|, \quad \forall \beta \leqslant \beta_0$.

The constant $\beta_0$ is sharp in the sense that for all $\beta>\beta_0$ there is a sequence of functions $u_k \in W_0^{1,n}(\Omega)$ satisfying $\displaystyle\int_\Omega {{{\left| {\nabla u_k} \right|}^n}dx} \leqslant 1$

but the integral $\displaystyle\int_\Omega {\exp (\beta {|u_k|^\frac{n}{n-1}})dx}$

grow without bound.

For general compact closed manifold $(M,g)$ the constant on the right hand side of the Moser-Trudinger inequality depends on the metric $g$. Working on a sphere $(\mathbb S^2,g_c)$ with a canonical metric allows us to control the constants.

Theorem (Moser). There is a universal constant $C_1>0$ such that for all $u \in W^{1,2}(\mathbb S^2)$ with $\displaystyle\int_{\mathbb S^2} {{{\left| {\nabla u} \right|}^n}dv_{g_c}} \leqslant 1$

and $\displaystyle\int_{\mathbb S^2} u dv_{g_c}=0$

we have $\displaystyle\int_{\mathbb S^2} \exp(4\pi u^2) \leqslant {C_1}$.

Observe that $\displaystyle 4\pi = \int_{{\mathbb{S}^2}} {d{v_{{g_c}}}} <\int_{{\mathbb{S}^2}} {{e^{4\pi {u^2}}}d{v_{{g_c}}}} \leqslant {C_1}$.

In the same way as we introduce in the entry concerning the Trudinger inequality one can show

Corollary. For $\displaystyle C_2 := \log C_1 +\log\frac{1}{4\pi}$

one has $\displaystyle \log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_2}$

for all $u \in W^{1,2}(\mathbb S^2)$.

Obviously, $C_2 >0$ since $C_1 >4\pi$. It turns out to determine the best constant $C_2$. This had been done by Onofri known as the Onofri inequality [here].

Theorem (Onofri).Let $u \in W^{1,2}(\mathbb S^2)$ then we have $\displaystyle\log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u$

with the equality iff $\Delta u +e^{2u}=1$.

The proof of the Onofri inequality relies on a result due to Aubin

Theorem (Aubin). For all $\varepsilon>0$ there exists a constant $C_\varepsilon$ such that $\displaystyle\log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\left( {\frac{1}{2} + \varepsilon } \right)\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_\varepsilon }$

for any $u$ belonging to the following class $\displaystyle S = \left\{ {u \in {W^{1,2}}({\mathbb{S}^2}):\int_{{\mathbb{S}^2}} {{e^{2u}}{x_j}d{v_{{g_c}}}} = 0,j = \overline {1,3} } \right\}$.

Source: S-Y.A. Chang, Non-linear elliptic equations in conformal geometry, EMS, 2004.