TITLE: Solving Laplace's equation in a sphere with mixed boundary conditions on the surface.
QUESTION [5 upvotes]: Can anyone help point me to a solution method for this problem?
Solve $C(\vec{x})$, where $\vec{x}=(r,\theta,\phi)$ on
$\Omega=\{\vec{x}\in\mathbb{R}^3\ |\ r\in[0,R],\ \phi\in[0,2\pi),\ \theta\in[0,\pi)\}$, where $R>0$. We define the boundaries and regions within $\Omega$ as follows:
\begin{align}
\partial\Omega_1 &=
%
\{\vec{x}\in\mathbb{R}^3\ |\ r=R,\ \theta\in[0,\theta_1),\ \phi\in[0,2\pi)\}\\
%
\partial\Omega_2 &=
%
\{\vec{x}\in\mathbb{R}^3\ |\ r=R,\ \theta\in[\theta_1,\theta_2),\ \phi\in[0,2\pi)\}\\
%
\partial\Omega_3 &=
%
\{\vec{x}\in\mathbb{R}^3\ |\ r=R,\ \theta\in[\theta_2,\pi),\ \phi\in[0,2\pi)\}
\end{align}
$C(\vec{x})$ is governed by the diffusion equation within $\Omega$ with boundary conditions given below,
\begin{align}
%
0 &= \nabla^2 C
%
\qquad &\text{for}\ \vec{x}\in\Omega \\
%
-\vec{n}\cdot\nabla C &= -\mu
%
\qquad &\text{for}\ \vec{x}\in\partial\Omega_1\\
%
-\vec{n}\cdot\nabla C &= \sigma C
%
\qquad &\text{for}\ \vec{x}\in\partial\Omega_2\\
%
-\vec{n}\cdot\nabla C &= 0
%
\qquad &\text{for}\ \vec{x}\in\partial\Omega_3
\end{align}
where $\mu,\sigma>0$.
By symmetry the problem reduces to
\begin{align} 
0 =&
%
\frac{\partial }{\partial r}\left(
r^2
\frac{\partial C}{\partial r}
\right)
%
+
\frac{1}{\sin{\theta}}
\frac{\partial}{\partial \theta}
\left(
\sin{\theta}
\frac{\partial C}{\partial \theta}
\right)
\end{align}
With the same BC, however I can't find a solution method that does not cause the problem to become badly posed.

EDIT: I have come across this paper by Mottin, I am unsure of its applicability here due to the piecewise definition of our Robin boundary condition. Does this invalidate the result of this paper?

REPLY [2 votes]: The paper [Mottin,2016] corresponds to the case where the boundaries are the pure Robin conditions (h is a constant).
For your boundary conditions see the paragraph 8.3 of this paper and the references:
[Alessandrini G. , Piero L. D. , Rondi L., Stable determination of corrosion by a single electrostatic boundary measurement, Inverse Probl. 2003; 19:973-984.]
[Fasino D, Inglese G. An inverse Robin problem for Laplace’s equation: theoretical results and numerical methods. Inverse Probl. 1999;15:41–48].