Well, I have the problem: $\Delta u(x,y)=0 \text{ on } \Omega_{a}=\{(x,y)\in\mathbb{R}^{2}:a<\lVert (x,y)\rVert<1\}$

$u(x,y)=1 \text{ on } \lVert (x,y)\rVert=a$

$u(x,y)=0 \text{ on } \lVert (x,y)\rVert=1$

And the question, is there any solution of this that is not radial?

Well, my approach is that the domain is symmetric so as the Laplace operator is invariant by rotation we can obtain something but I don't see this very clear. And if this works, why in the problem on the $B(0,1)$ doesn't admit radial solutions?

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    $\begingroup$ Do you know how to prove uniqueness of solutions to Laplace's equation? $\endgroup$ – Chappers Jun 9 '17 at 19:09
  • $\begingroup$ I think so, studying the uniqueness of the problem with homogeneous boundary conditions, and with the maximum and minimum principle we can conclude that $u$ is 0, not? $\endgroup$ – Skullgreymon Jun 9 '17 at 19:24
  • $\begingroup$ Yes, that's the idea. So find a radial solution and show that it's unique. $\endgroup$ – Chappers Jun 9 '17 at 19:47
  • $\begingroup$ Thank you very much, I think I understand the solution, we can think on radial solutions because the 0 is not include on our domain and adjusting our parametres of the fundamental solution we can obtain the unique solution. $\endgroup$ – Skullgreymon Jun 9 '17 at 19:56
  • $\begingroup$ But one more question, this is possible because the domain is "radial" not? If the domain is a rectangle we can't find radial solutions not? $\endgroup$ – Skullgreymon Jun 9 '17 at 19:57

$u(r,\theta)=\frac{\log{r}}{\log{a}}$ is a radial solution. We now show that a solution to this problem is unique: let $u,v$ be two solutions. Then $w=u-v=0$ on both parts of the boundary and is harmonic. Then if $w \neq 0$ (so $u \neq v$) $$ 0 < \int_{\Omega_a} \lvert \nabla w \rvert^2 \, dV = \int_{\partial \Omega_a} w \nabla w \cdot d\mathbf{s} - \int_{\Omega_a} w\Delta w \, dV = 0, $$ since $w=0$ on the boundary and $\Delta w=0$ on $\Omega_a$, which is a contradiction unless $w=0$ everywhere, so $u=v$. Hence $u$ is the only solution.

There is a radial solution to Laplace's equation on $B(0,1)$ with the boundary condition $u(1,\theta)=0$, but it's exactly zero, by exactly the same argument (and indeed, this is the limit as $a\to 0$ of the $\Omega_a$ solution).

For a rectangle, the uniqueness result still holds, but the solution is instead made out of Fourier series (one can construct solutions that are nonzero only on one side of the rectangle and sum them, for example).


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