# Ngô Quốc Anh

## April 18, 2013

### A lower bound for solutions involving distance functions

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 21:14

In this note, we discuss an useful lemma and its beautiful proof given by Brezis and Cabré in a paper published in Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. in 1998. The full article can be freely downloaded from here. Before saying anything further, let us state the lemma.

Lemma 3.2. Suppose that $h \geq 0$ belongs to $L^\infty (\Omega)$. Let $v$ be the solution of

$\left\{\begin{array}{rcl} - \Delta v &=& h \quad \text{ in }\Omega , \hfill \\ v &=& 0 \quad \text{ on }\partial \Omega . \hfill \\ \end{array}\right.$

Then

$\displaystyle\frac{{v(x)}}{{\text{dist}(x,\partial \Omega )}} \geqslant c\int_\Omega {h\text{dist}(x,\partial \Omega )} ,\qquad\forall x \in \Omega,$

where $c>0$ is a constant depending only on $\Omega$.

This type of estimate frequently uses in the literature. We now show the proof of the lemma.