(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

From the book Evans-PDE, p.24, equation (12),

It is written that

[tex]C ||D^2f||_{L_{\infty}(R^n)} \int_{B(0,\epsilon)}|\Phi(y)|dy

\\ \leq \begin{cases} C \epsilon^2 |\log{}\epsilon| & (n=2) \\ C \epsilon^2 & (n \geq 3) \end{cases}[/tex]

How is this?

2. Relevant equations

[tex]\Phi(y) = \begin{cases} -\frac{1}{2\pi}\log{}|y| & (n=2) \\ \frac{1}{n(n-2)\alpha(n)} \frac{1}{|y|^{n-2}} & (n \geq 3) \end{cases}[/tex]

for [tex]y \in \mathbb{R}^n-0[/tex] (the fundamental solution of Laplace's equation).

[tex]\alpha(n)[/tex] is the volume of the unit ball in [tex]\mathbb{R}^n[/tex].

[tex]C[/tex] is a constant.

3. The attempt at a solution

Take n=3. Then,

[tex]\int_{B(0,\epsilon)}|\Phi(y)|dy = C\int_{0}^{\epsilon}\int_{0}^{2\pi}\int_{0}^{2\pi}\frac{1}{r} d\theta d\phi dr[/tex]

[tex]= C \int_{0}^{\epsilon}\frac{1}{r} dr[/tex]

[tex]= C (\log{}(\epsilon)-\log{}(0)) = \infty[/tex]

What am I doing wrong?

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# Homework Help: Evans-pde-laplace eqn

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