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.

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.

(more…)

September 14, 2010

Asympotic behavior of integrals, 2

Filed under: Giải Tích 6 (MA5205), PDEs — Tags: — Ngô Quốc Anh @ 15:07

We now prove the following result

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} .

(more…)

September 7, 2010

Asympotic behavior of integrals

Filed under: Giải Tích 6 (MA5205), PDEs — Tags: — Ngô Quốc Anh @ 10:52

Long time ago, we studied [here] the following fact

Suppose f \in L^1(\mathbb R^n) \cap L_{loc}^\infty (\mathbb R^n) with f \geq 0. Define

\displaystyle Sf\left( x \right) = \int_{\mathbb{R}^n } {\log \frac{{\left| y \right|}}{{\left| {x - y} \right|}}f\left( y \right)dy}.

Show that Sf(x) is finite for all x \in \mathbb R^n and Sf \in L_{loc}^1(\mathbb R^n).

In this entry, from now on we continue to prove several useful results appearing in PDE. We shall prove the following

Theorem. Assume u is a solution to

\displaystyle (-\Delta)^\frac{3}{2} u(x)=-2e^{3u(x)}, \quad x \in \mathbb R^3

with finite energy

\displaystyle \int_{{\mathbb{R}^3}} {{e^{3u(x)}}dx} < \infty.

Then

\displaystyle\mathop {\lim }\limits_{|x| \to \infty } \frac{{u(x)}}{{\log |x|}} = - \frac{1}{{{\pi ^2}}}\int_{{\mathbb{R}^3}} {{e^{3u(y)}}dy} .

(more…)

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