I am currently reading Evan's PDE and am getting hung up on many of the more "technical details". This question may be very basic (multivariable calculus). I am given that the fundamental solution of Laplace's equation is $$ \Phi(x) := \begin{cases} -\frac{1}{2 \pi} \log |x| & (n=2) \\
\frac{1}{n(n-2) \alpha(n)} \frac{1}{|x|^{n-2}} & (n \ge 3) \end{cases}$$
How would I evaluate $$ \int_{B(0, \epsilon)} |\Phi(y) | dy ? $$
Best Answer
Change to polar coordinates?
For $n \geq 3$, note
$$ \int_{B(0,\epsilon)} \frac{1}{|x|^{n-2}} \mathrm{d}x = C_n \int_0^\epsilon r^{2-n} r^{n-1} \mathrm{d}r = C_n \int_0^\epsilon r \mathrm{d}r = \frac{1}{2} C_n \epsilon^2 $$ where $C_n$ is the are of the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$.
For $n = 2$ you need to integrate $\int_0^\epsilon r \log(r)\mathrm{d}r$ which can be evaluated using integration by parts.