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.

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