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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 $f_2=2 |x|$

  1. Show that $|u_1 (x,y)- u_2 (x,y)| \leq 1$

  2. 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

Andrew142
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  • You should try to write down what you've tried or did not understand, the goal is not for us to do your homework – julio_es_sui_glace Jun 25 '23 at 17:52
  • Well 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 – Andrew142 Jun 25 '23 at 18:05
  • Edit your post then :) – julio_es_sui_glace Jun 25 '23 at 18:07
  • For 2 use the maximum principle to deduce that $u_1\equiv 1$ and that $u_2$ takes its minimum and maximum on $\partial S,.$ That minimum and maximum are then which values ? Rest should be easy. – Kurt G. Jun 25 '23 at 20:56

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