1. ## Legendre Polynomial problem

Find $\theta \ \ s.t. \ \ \Delta^{2}\phi=0, \ \ a

2. Originally Posted by silversand
Find $\theta \ \ s.t. \ \ \Delta^{2}\phi=0, \ \ a
Not clear, provide context and explain notation please.

CB

3. Originally Posted by silversand
Find $\theta \ \ s.t. \ \ \Delta^{2}\phi=0, \ \ a
I'm guessing what you mean is find $\phi$ not $\theta$. From your BC, it's desirable to switch to polar coords, so your PDE becomes

$
\phi_{rr} + \frac{\phi_r}{r} + \frac{\phi_{\theta \theta}}{r^2} = 0
$
with your BC $\phi(a,\theta) = \phi(b,\theta) = \frac{1}{2} - \frac{\cos 2 \theta}{2}$

We'll shift the BC a bit by introducing $\psi = \phi - \frac{1}{2}$ so the PDE is the same but the new BCs is $\psi(a,\theta) = \psi(b,\theta) = - \frac{\cos 2 \theta}{2}$. Fortunately for us, this problem admits separable solutions so

$
\psi = \left( c_1 r^2 + \frac{c_2}{r^2}\right) \left( A \sin 2 \theta + B \cos 2 \theta \right)
$

Applying the boundary condtions gives us (we can set $B = 1$ wlog)

$
c_1 = - \frac{1}{2(a^2+b^2)},\;\;\; c_2 = -\frac{1}{2} \frac{a^2b^2}{a^2+b^2},
$

which leads to the final solution

$
\phi = \frac{1}{2} - \frac{1}{2(a^2+b^2)} \left(r^2 + \frac{a^2b^2}{r^2} \right)\cos 2 \theta.
$

Side note: Title says Legendre polynomial - not sure why? They typically arise in 3D geometries but nothing in this post suggest that the problem is in $\mathbb R^3$.