Ngô Quốc Anh

March 26, 2011

Asympotic behavior of integrals, 4

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 21:25

We consider the following PDE

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

By letting

$\displaystyle w(x) = \frac{1}{{2\pi }}\int_{{\mathbb{R}^2}} {\left[ {\log |x - y| - \log |y|} \right]f(y)dy}$

via the potential theory, we has already proved that

$u-w={\rm const.}.$

As such, the analysis of $w$ turns out to be the core of the studying of solutions to our PDE. As in this entry, we showed that the following limit

$\displaystyle\mathop {\lim }\limits_{|x| \to \infty } \left[ {w(x) - \alpha \log |x|} \right] = -\frac{1}{{2\pi }}\int_{{\mathbb{R}^2}} {\log |y|f(y)dy}$

exists for certain function $f$. Not just the behavior at the infinity, as a question proposed also in that entry, we can control the decay rate of

$\displaystyle {w(x) - \alpha \log |x| + \frac{1}{{2\pi }}\int_{{\mathbb{R}^2}} {\log |y|f(y)dy} }$

i.e. we need the fact

$\displaystyle\left| {w(x) - \alpha \log |x| + \frac{1}{{2\pi }}\int_{{\mathbb{R}^2}} {\log |y|f(y)dy} } \right| \leqslant \frac{{C\log |x|}}{{|x|}},\quad \forall |x| \geqslant 1$

for some positive constant $C$ where $w$ is a particular solution to

I do think this result is correct since it has been used once in a paper by X.X. Chen published in Calc. Var. Partial Differential Equations [here] but some idea is involved. I leave here as my own open question needed to be addressed in the future.