The Sobolev norm of characteristic functions with applications to the Calderón inverse problem

Abstract

We consider Calderón's inverse problem on planar domains Ω with conductivities in fractional Sobolev spaces. When Ω is Lipschitz, the problem was shown to be stable in the L2-sense in Clop et al. [Stability of calderón's inverse conductivity problem in the plane for discontinuous conductivities, Inverse Probl. Imaging 4 (2010), 49–91]. We remove the Lipschitz condition on the boundary. To this end, we analyse the Sobolev regularity of the characteristic function of Ω. For Ω a quasiball, we compute ||χΩ||Ws,p(ℝd) in terms of the δ-neighbourhoods of the boundary.