Calderón’s inverse conductivity problem in the plane. (English) Zbl 1111.35004

Astala, Kari; Päivärinta, Lassi

Ann. Math. (2) 163, No. 1, 265-299 (2006).

Summary: We show that the Dirichlet to Neumann map for the equation \(\nabla\cdot \sigma\nabla u=0\) in a two-dimensional domain uniquely determines the bounded measurable conductivity \(\sigma\). This gives a positive answer to a question of A. P. Calderón from 1980. Earlier the result has been shown only for conductivities that are sufficiently smooth. In higher dimensions the problem remains open.

MSC:

35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
30C62	Quasiconformal mappings in the complex plane
35R30	Inverse problems for PDEs
35J25	Boundary value problems for second-order elliptic equations
35R05	PDEs with low regular coefficients and/or low regular data

Keywords:

Dirichlet to Neumann map; bounded measurable conductivity