I'm currently working on my master's thesis, but I'm stuck.
Consider the PDE $\nabla\cdot (\sigma\nabla u)=0$ on $B_1\subset \mathbb{R^3}$, where $\sigma:B_1\rightarrow \mathbb{R}_+$ is constant inbetween the radii $0<r_1...<r_N<1$. Assume boundary conditions $(\sigma\nabla u)\cdot n=g$. I would like $g$ to be a sum of dirac measures, but unfortunatly this isn't well defined for $H^\frac{1}{2}(S^2)$, so assume $g\in (H^{\frac{1}{2}}(S^2))^\ast$ (or maybe $L^2$, I'm still unsure), with $g(1_{S^2})=0$.
So we get the weak problem: $$\int_{B_1}(\sigma\nabla u) \cdot \nabla \phi=g(\phi).$$
By the theorem of Lax Milgram we get the existence of a solution in $H^1(B_1)$, which is unique up to a constant.
I want to compare this to my FEM solution, therefore I would like to find a (semi-)analytic expression of $u$. In a lot of books, mainly directed at physicists, it is stated that the solution to the laplace equation has the form $$u(r,\theta,\phi)=\sum_{l=0}^\infty\sum_{m=-l}^l (A_{l,m}r^l+B_{l,m}r^{-l-1})Y_l^m(\theta,\phi).$$
Unfortunately, I can't find anything mathematically rigorous about this, e.g. convergence in $H^1$ or compatibility with the boundary conditions. I could then use this to construct my solution shell by shell.
Let me further note, that I'm also looking at a similar PDE with different boundary conditions given by $u+Z_l(\sigma\nabla u)\cdot n=U_l$ on $\Gamma_l\subset S^2$ with $\int_{\Gamma_l}(\sigma\nabla u)\cdot n=I_l$ for $l=1..L$ and $(\sigma\nabla u)\cdot n=0$ on the rest of the boundary. Here the solution consists of a pair $(u,U)\in H^1\oplus\mathbb{R}^L$. I would like to find $u$ as it is done in [1] for the 2D case, but again they use a similar expression for $u$ to the one above just for the 2D case.
Every help is welcome. Thank you very much.
[1] Somersalo, Erkki, et al. “Existence and Uniqueness for Electrode Models for Electric Current Computed Tomography.” SIAM Journal on Applied Mathematics, vol. 52, no. 4, Society for Industrial and Applied Mathematics, 1992, pp. 1023–40, http://www.jstor.org/stable/2102189.
EDIT:
Since $H^{\frac{1}{2}}(S^2)$ is a Hilbert space, Riesz theorem yields a function in $H^{\frac{1}{2}}(S^2)$, which we will call again $g$, with $g(\phi)=\langle g, \phi\rangle_{H^\frac{1}{2}}$. We can express $g$ in spherical harmonics $g=\sum a_{l,m}Y_l^m$ and I think the $H^\frac{1}{2}$ norm is given by a weighted sum of these coefficients (I still need to find a reference/proof of this) and we furthermore have convergence in this norm. Plugging in the series gives $\sigma \nabla u \cdot n|_{S^2}=\sigma\sum_{l,m}( l A_{l,m}-(l+1) B_{l,m})Y_l^m $ and comparing coefficients yields $\sigma|_{S^2}(l A_{l,m}-(l+1)B_{l,m})=a_{l,m}$. At the "boundary" $B_{r_N}$ we get some condtions by restricting $u$. Therefore we get a system of equations that determine $A_{l,m}$ and $B_{l,m}$. Similar we can derive the coefficients for the other shells by the conditions that $u$ and $\sigma \nabla u \cdot n$ are continuous at the shell boundaries. But then I'm stuck with the $H^1$ convergence.
The $u$ constructed above unfortunately does not satisfy the boundary conditions. We would get $$\int_{B_1}(\sigma\nabla u) \cdot \nabla \phi=\int_{S^2}g \phi\not = \langle g,\phi \rangle_{H^\frac{1}{2}}.$$
Next try:
On $H^\frac{1}{2}(S^2)$ I was using the norm $\Vert v\Vert^2_{H^\frac{1}{2}}=\Vert v\Vert^2_{L^2}+\vert v\vert^2_{H^\frac{1}{2}} $ with the Gagliardo semi-norm. In this paper they use $\Vert v\Vert^2_{H^\frac{1}{2}}=\Vert v\Vert^2_{L^2}+\Vert (-\Delta_S)^\frac{1}{4}v\Vert^2_{L^2}.$ I'm not familiar with the Laplace–Beltrami operator but I think this could be equivalent. If you know a reference or something please let me know. In the paper they state, that for $f=\sum_{l,m}f_{l,m}Y_l^m\in L^2(S^2)$ we get $$\Vert f\Vert^2_{H^\frac{1}{2}(S^2)}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}(1+l^\frac{1}{2}(l+1)^\frac{1}{2})\vert f_{l,m}\vert^2.$$ Notice that this satisfies the parallelogram law, and we see that the spherical harmonics are orthogonal wrt to this inner product. Now we want to find $\tilde{g}\in L^2(S^2)$ with $$\int_{S^2}\tilde{g} \phi = \langle g,\phi \rangle_{H^\frac{1}{2}}$$ for $\phi\in H^\frac{1}{2}$. Then $\sigma\nabla u\cdot n=\tilde{g}$. Expressing everything in spherical harmonics and setting $\phi=Y_l^m$ yields $$\tilde{g}_{l,m}=\int_{S^2}\tilde{g} Y_l^m=\langle g,Y_l^m \rangle_{H^\frac{1}{2}}=g_{l,m}(1+l^\frac{1}{2}(l+1)^\frac{1}{2})$$ where the last part may has to be changed, depending on how the two inner products relate to each other. I'm uncertain about this. Since $g\in H^\frac{1}{2}$ we get $\sum\vert g_{l,m}\vert^2(1+l^\frac{1}{2}(l+1)^\frac{1}{2}) <\infty$ but I cannot conclude $\tilde{g}\in L^2$.
There was a fallacy in my thinking. Such a function $\tilde{g}$ must not exist in $L^2$. I think correct would be the following: Let us set $u_{l,m}(r)=A_{l,m}r^l+B_{l,m}r^{-l-1}$ $$\langle g, \phi\rangle_{H^\frac{1}{2}}=\sum_{l,m}\int_{B_1}\sigma \nabla (u_{l,m}Y_l^m)\cdot \nabla \phi=\sum_{l,m}\int_{S^2}\sigma \frac{\partial u_{l,m}}{\partial r}(1)Y_l^m \phi$$ So we were just looking for the sequence $\tilde{g}_{l,m}$ given by $$g_{l,m}(1+l^\frac{1}{2}(l+1)^\frac{1}{2})= \tilde{g}_{l,m}=\sigma \frac{\partial u_{l,m}}{\partial r}(1)=\sigma( l A_{l,m}-(l+1) B_{l,m}).$$ This should satisfy the boundary conditions. So only the problem with the convergence remains.
Since we know that there already exists a solution $u\in H^1$ which satiesfies the boundary conditions and the PDE, it is enough to show that $\tilde{u}=\sum_{l=0}^\infty\sum_{m=-l}^l (A_{l,m}r^l+B_{l,m}r^{-l-1})Y_l^m$ solves the laplace equation in one layer together with Dirichlet boundary conditions given by the restriction of $u$ to the layer boundaries (since the solution to this problem is also unique). When expressing these restrictions in spherical harmonics, e.g. $u(R_1,\cdot, \cdot)=\sum_{l,m}a_{l,m}Y_l^m$ and $u(R_2,\cdot ,\cdot)=\sum_{l,m}b_{l,m}Y_l^m$, we can express the coefficients $A_{l,m}, B_{l,m}$ in terms of $a_{l,m}, b_{l,m}$. Since the restrictions of $u$ are in $H^\frac{1}{2}$ we get $(1+l^\frac{1}{2}(l+1)^\frac{1}{2})^\frac{1}{2}a_{l,m}\in \ell^2$ and the same for $b_{l,m}$. With this we can show the convergence in $H^1(B_{R_1}\backslash B_{R_2})$ on this layer. Putting all together we get the desired convergence. This paper helps to deal with the gradients of the spherical harmonics.