Ngô Quốc Anh

February 26, 2011

Decay estimates of a linear problem in R^2

Filed under: PDEs — Ngô Quốc Anh @ 2:36

I found the following simple result in a paper due to F. Lin and J.C. Wei published in Commun. Pure Applied Math. this year [here].

Lemma 4.2. Let $h$ satisfy $-\Delta h=f(z), \quad h(\overline z)=-h(z), \quad |h| \leqslant C$

where $f$ satisfies $\displaystyle |f(z)| \leqslant \frac{C}{(1+|z|)^{2+\sigma}},\quad 0<\sigma<1.$

Then $\displaystyle |h(z)| \leqslant \frac{C}{(1+|z|)^{\sigma}}$.

Proof.

By the Poisson formula, $\displaystyle h(z)=\frac{1}{2\pi}\int_{\{y_2 >0\}}\log \frac{|\overline z - y|}{|z-y|}f(y)dy.$

Because of the grown of $f$, it is known that $h(z) \to 0$ as $|z| \to +\infty$. We construct suitable supersolutions on $\{x_2 > 0\}$. Then the result will follow from the maximum principle.

In fact, let $h_0(z)=r^\beta x_2^\gamma$

where $r=|z|$ and the parameters are chosen so that $\beta+\gamma=-\sigma, \quad 0<\sigma<\gamma<1$.

Then simple computations show that $\displaystyle\begin{gathered} \Delta {h_0} = \Delta ({r^\beta }x_2^\gamma ) \hfill \\ \qquad= {r^\beta }x_2^\gamma \left[ {({\beta ^2} + 2\beta \gamma ){r^{ - 2}} + \gamma (\gamma - 1)x_2^{ - 2}} \right] \hfill \\ \qquad\leqslant - C{r^\beta }x_2^\gamma ({r^{ - 2}} + x_2^{ - 2}) \hfill \\ \qquad\leqslant - C{r^{\beta - 1}}x_2^{\gamma - 1} \hfill \\ \qquad\leqslant - C{r^{\beta + \gamma - 2}} \hfill \\ \qquad\leqslant - C{(1 + |z|)^{\beta + \gamma - 2}}. \hfill \\ \end{gathered}$