# Ngô Quốc Anh

## December 31, 2011

### A Hardy-Moser-Trudinger inequality: A conjecture by Wang and Ye

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 21:48

Let $B$ denote the standard unit disk in $\mathbb R^2$. The famous Moser–Trudinger inequality says that

$\displaystyle\int_B {\exp \left( {\frac{{4\pi {u^2}}}{{\left\| {\nabla u} \right\|_2^2}}} \right)dx} \leqslant C < \infty ,\quad\forall u \in H_0^1(B)\backslash \{ 0\}$

holds. There is another important inequality in analysis, the Hardy inequality which claims that

$\displaystyle H(u) = \int_B {|\nabla u{|^2}dx} - \int_B {\frac{{{u^2}}}{{{{(1 - |x{|^2})}^2}}}dx} \geqslant 0,\quad\forall u \in H_0^1(B)$

holds. The one $H$ is usuall called the Hardy functional. One can immediately see that

$\displaystyle\frac{{4\pi {u^2}}}{{\left\| {\nabla u} \right\|_2^2}} \leqslant \dfrac{{4\pi {u^2}}}{{\displaystyle\int_B {|\nabla u{|^2}dx} - \int_B {\frac{{{u^2}}}{{{{(1 - |x{|^2})}^2}}}dx} }}$

for any $u \in H_0^1(B)\backslash \{ 0\}$. Recently, in a paper accepted in Advances in Mathematics journal, Wang and Ye proved that there exists a constant $C_0 >0$ such that the following

$\displaystyle\int_B {\frac{{4\pi {u^2}}}{{H(u)}}dx} \leqslant C_0 < \infty ,\quad\forall u \in \mathcal H(B^n)\backslash \{ 0\}$

where $B^n$ is the unit ball in $\mathbb R^n$, $n \geqslant 2$ and $\mathcal H=\mathcal H(B^n)$ is the complement of $C_0^\infty(B^n)$ with respect to the following norm $\|u\|_{\mathcal H}=\sqrt{H(u)}$.

Let us go back to the case $n=2$. They then defined

$\displaystyle {H_d}(u) = \int_\Omega {|\nabla u{|^2}dx} - \frac{1}{4}\int_\Omega {\frac{{{u^2}}}{{d{{(x,\partial \Omega )}^2}}}dx} > 0,\quad \forall u \in H_0^1(\Omega )\backslash \{ 0\}$

where $\Omega$ is a regular, bounded and convex domain sitting in $\mathbb R^2$. They then conjectured that the following

$\displaystyle\int_\Omega {\frac{{4\pi {u^2}}}{{{H_d}(u)}}dx} \leqslant C(\Omega ) < \infty ,\quad\forall u \in {\mathcal H_d}(\Omega )\backslash \{ 0\}$

still holds for some constant $C(\Omega)>0$ where ${\mathcal H_d}(\Omega )$ denotes the completion of $C_0^\infty (\Omega)$ with the corresponding norm associated with $H_d$. Apparently, the conjecture holds true for $\Omega = B$.