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.

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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?

  • $\begingroup$ what do you mean by "badly posed"? $\endgroup$ – Arashium Feb 28 '16 at 0:22
  • $\begingroup$ @Arashium: For example one of my attempts to segment the region and then match regions ended with the coefficients being over defined. $\endgroup$ – Freeman Feb 29 '16 at 17:34

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].


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