Let $B$ the unit disk in $\mathbb{R}^{2}.$ Let $B_{\epsilon}$ be the disk centered at the origin at of radius $\epsilon.$ Let me define $\displaystyle \sigma=\mathcal{I}_{B}+\mathcal{I}_{B_{\epsilon}}$ (each of this is the characteristic function of the indexed set).
I have the Dirichlet problem:
$\text{div}\sigma \nabla u=0, \hspace{3mm} \text{in} \hspace{1mm} B $
$u=f, \hspace{3mm} \text{in} \hspace{1mm} \partial B,$
With $f$ an $H^{1/2}(\partial B)$ function.
I know this problem is well-posed (this is not hard work). What I want to do is to find an explicit solution, represented by a Fourier-type series, given a certain $f.$
Basically, I have tried testing against function supported in a small ball near the discontinuity in order to obtain transmission conditions, but I got nothing. Any help would be appreciated.
REMARK: Some motivation: I am interested in computing explicitly $\dfrac{\partial u}{\partial \nu} \vert _{\partial B}.$ The idea is that if I do so I could prove that stability in the inverse conductivity problem (the Calderón's problem) is impossible if we do not assume any smoothness at all on the conductivities. I understand the counterexample is originally Alesandrini's, but I have not found this explained anywhere. References are welcomed.
Thanks