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.


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