# Ngô Quốc Anh

## September 18, 2010

### Asympotic behavior of integrals, 3

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 11:56

In the previous entry, we showed the following

Theorem. Let $u$ and $f$ be two smooth functions on $\mathbb R^2$ satisfying

$\Delta u(x)=f(x), \quad x \in \mathbb R^2$.

Suppose that $f$ is bounded and also $f \in L^1(\mathbb R^2)$ and

$|u(x)| \leqslant o(|x|), \quad |x| \to \infty$.

Then

$\displaystyle\mathop {\lim }\limits_{|x| \to \infty } \frac{{u(x)}}{{\log |x|}} = \frac{1}{{2\pi }}\int_{{\mathbb{R}^2}} {f(y)dy}$.

As suggested in an earlier entry, in this topic, we show that the following limit

$\displaystyle\mathop {\lim }\limits_{|x| \to \infty } \left[ {u(x) - \alpha \log |x|} \right]$

exists where

$\displaystyle\alpha = \frac{1}{{2\pi }}\int_{{\mathbb{R}^2}} {f(y)dy}$

for some good $f$.