I am trying to solve this boundary value problem in three dimensions:
$$L(u)= 2rcos(\theta)-1$$ inside of unit ball ($r<1$) $$\frac{\partial u}{\partial r}= 3u$$ on the boundary ($r=1$)
where L(u) is Laplace operator.
I should solve it using Legendre polynomials but I do not know how to aproach it. The second question is if there is only one bounded solution to this problem.
Thank you for your help.
Edit:
I use this spherical coordinates: $x=rsin(\theta)cos(\phi), y=rsin(\theta)sin(\phi), z=rcos(\theta)$.
So the problem is formulated in these coordinates. I want to solve this using Legendre polynomials to somehow prove it has only one unique bounded solution or it has more than one solution that is bounded.
you first need to clear your mind, first you have the equation of Laplacian equals something using polar coordinates I guess. So, the problem is badly formulated, it should be $\Delta u = 2z-1$. A problem like this we solve by trying to guess "particular solution" (one which kills 2z-1 on the other side) and you should try with something as $u(x,y,z) = A(x^2+y^2+z^2)z+ B(x^2+y^2+z^2)$ with appropriate $A$ and $B$, so you have $u_{xx} = 2Az+2B,\ u_{yy} = 2Az+2B,\ u_{zz} = 6Az + 2B$, so $\delta u = 10Az + 6B = 2z-1$ leading to $A = \frac{1}{5}$, $B=\frac{-1}{6}$.
Now,consider a shift $u(x,y) = v(x,y) + \frac{1}{5}z(x^2+y^2+z^2)-\frac{1}{6}(x^2+y^2+z^2)$, now at a boundary you have $x^2+y^2+z^2=1$, so you have $u(x,y,z) = \frac{1}{5}z - \frac{1}{6}+v(x,y,z)$, so $u(\phi, \theta, r) = \frac{1}{5}r\cos(\theta) - \frac{1}{6}+v(\phi, \theta, r)$, so $u_r = \frac{1}{5}\cos(\theta)+v_r(\phi,\theta, r)$ so $\frac{1}{5}\cos(\theta)+v_r(\phi,\theta, r)=\frac{3}{5} \cos(\theta) + 3v(\phi,\theta, r)$ from which we see $v_r=3v+\frac{2}{5}\cos(\theta)$ on the boundary.
So now you have $\delta v = 0$ and $v_r=3v+\frac{2}{5}\cos(\theta)$. What you now need is a Laplacian in polar coordinates in three dimensions and then you look for solution in the shape $v(x,y,z) = R(x)T(y)C(z)$ and discuss what those three functions are, this would be a five-pages novel, I think you can solve it no problems.