# Ngô Quốc Anh

## August 2, 2010

### The Brezis-Merle inequality

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 0:25

I am going to talk about uniform estimates and blow-up phenomena for solutions of

$\displaystyle -\Delta u=V(x)e^u$

in two dimensions done by Brezis and Merle around 1991 published in Comm. Partial Differential Equations [here]. As a first step, I am going to derive some inequality that we need later.

Assume $\Omega \subset \mathbb R^2$ is bounded domain and let $u$ be a solution of

$\displaystyle -\Delta u=f(x)$

together with Dirichlet boundary condition. Here function $f$ is assumed to be of class $L^1(\Omega)$.

Theorem (Brezis-Merle). For every $\delta \in (0,4\pi)$ we have

$\displaystyle\int_\Omega {\exp \left[ {\frac{{(4\pi - \delta )|u(x)|}}{{{{\left\| f \right\|}_1}}}} \right]dx} \leqslant \frac{{4{\pi ^2}}}{\delta }{\rm diam}{(\Omega )^2}$

where $\|\cdot\|_1$ denotes the $L^1$-norm and $u$ a solution to our PDE.