# Ngô Quốc Anh

## August 16, 2010

### The Moser-Trudinger inequality for domains with holes

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 2:42

In this entry, we are interested in the following result

Theorem (Moser-Trudinger’s inequality for domains with holes). Let $\Omega$ be a bounded smooth domain in $\mathbb R^2$. Let $S_1$ and $S_2$ be two subsets of $\overline \Omega$ satisfying

${\rm dist}(S_1,S_2) \geqslant \delta_0>0$

and let $\gamma_0$ be a number satisfying $\gamma_0 \in \left(0,\frac{1}{2}\right)$. Then for any $\varepsilon>0$, there exists a constant $c=c(\varepsilon, \delta_0, \gamma_0)>0$ such that

$\displaystyle\int_\Omega {{e^u}} \leqslant C\exp \left[ {\frac{1}{{32\pi - \varepsilon }}\int_\Omega {{{\left| {\nabla u} \right|}^2}} + C} \right]$

holds for all $u \in H_0^1(\Omega)$ satisfying

$\displaystyle\frac{{\int_{{S_1}} {{e^u}} }}{{\int_\Omega {{e^u}} }} \geqslant {\gamma _0}, \quad \frac{{\int_{{S_2}} {{e^u}} }}{{\int_\Omega {{e^u}} }} \geqslant {\gamma _0}$.