The analysis of the boundary value problems of EIT: in which spaces they are solvable, what the solution operators look like, and which structure the boundary data carry.
Reading order.
- Function spaces: Sobolev and Trace Spaces.
- Solvability: Weak Formulation of the Conductivity Equation, Green’s Identities, Lax-Milgram Theorem, with the Dirichlet Problem and the Neumann Problem as the two basic cases.
- Variational characterisation: Dirichlet and Thomson Principles.
- Boundary operators: Dirichlet-to-Neumann Map, Neumann-to-Dirichlet Map, Properties of the Boundary Operators, and the Forward Map from conductivities to data.
- Geometry: Conformal Invariance of the Conductivity Equation relates problems on different planar domains.
Next. Whether the data determine the conductivity is the subject of The Inverse Problem; the discrete versions of these operators are in Finite Elements.
References
- M. Cheney, D. Isaacson, J. C. Newell (1999). Electrical Impedance Tomography. SIAM Rev. 41(1), 85–101. doi:10.1137/S0036144598333613
- O. Steinbach (2008). Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer. doi:10.1007/978-0-387-68805-3