For the 2-dim Laplace equation on unit circle, Robin condition is given by $$u_r(1,\theta)-u(1,\theta)=\beta(\theta).$$ How to give a counter example to show the solution is not unique?

By the Laplace equation, I find $$u(r,\theta)=a_0+\sum_{n=1}^{\infty}r^n(a_n\cos(n\theta)+b_n\sin(n\theta)),$$ and plug into the robin condition we need to have $$\beta(\theta)=-a_0+\sum_{n=2}^{\infty}(n-1)(a_n\cos(n\theta)+b_n\sin(n\theta)).$$

How to create $\beta$ and $a_n, b_n$ so that the solution is not unique? Thank you!

  • $\begingroup$ When the coefficient is negative it is also called Steklov boundary condition. $\endgroup$ – timur Oct 31 '13 at 15:18
  • $\begingroup$ OK, thank you for your info! I haven't heard of this before. Thank you! $\endgroup$ – breezeintopl Oct 31 '13 at 15:22

Hint: what does your equation say about $a_1$ and $b_1$?

  • $\begingroup$ Thank you for your hint! Do you mean if I set $\beta=a\cos(\theta)+b\sin(\theta)$. In this way, we could have one coefficients $0,a,b,0,0,...$ and another coefficients $a_0,0,0,a_2,b_2,...$? $\endgroup$ – breezeintopl Oct 31 '13 at 15:09
  • $\begingroup$ Oh, it seems I got it. The Robin condition does not say any thing about $a_1,b_1$, so they two can be any number and thus not unique. Thank you! $\endgroup$ – breezeintopl Oct 31 '13 at 15:16

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