Let $S (0,0) \subset \mathbb{R}^2 $ be the circular disc with radius $r=1$ and center $(0,0)$. Furthermore let $u_1(x,y)$ and $u_2(x,y)$ be solutions of:
- $\Delta u_n =0 \quad (x,y) \in S(0,0)$
- $u_n =f_n \quad (x,y) \in \partial S(0,0)$
Where $n=1,2$, $f_1(x,y)=1$ and $f_2=2 |x|$
Show that $|u_1 (x,y)- u_2 (x,y)| \leq 1$
Calculate $u_1(0,0)-u_2(0,0)$
For 2. I used the mean value property to compute $u_1(0,0)$ and $u_2(0,0)$: $\frac{1}{2\pi} \int_{0}^{2\pi} 1dt - \frac{1}{2\pi} \int_{0}^{2\pi} 2 |cos(t)|dt=1-\frac{4}{\pi}$
I find 1. to be harder because I don't know how to solve the laplace equation on a circle, can someone help please