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.

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