Reconstruction of Rough Conductivities from Boundary Measurements
Ashwin Tarikere
TL;DR
This work extends Nachman’s constructive reconstruction to conductivities with limited regularity by proving reconstruction for $\gamma\in W^{3/2,2n}(\Omega)$ with $\gamma\equiv 1$ near the boundary and establishing a log-type stability when $\gamma\in W^{2-s,n/s}(\Omega)$ for $0<s<1/2$. The authors reduce the inverse conductivity problem to a Schrödinger problem with $q=\Delta\sqrt{\gamma}/\sqrt{\gamma}$, develop a Sobolev- and weight-based CGO framework at low regularity, and formulate a Fredholm boundary-integral equation that yields a constructive reconstruction of $q$ (and thus $\sqrt{\gamma}$) from the Dirichlet-to-Neumann map. They provide explicit stability results: a log-type estimate for $q$ in $W^{-1/2,2n}_{\mathrm{comp}}(\Omega)$ and a corresponding stability for $\gamma$ in a Hölder class, under Sobolev regularity assumptions that are weaker than Lipschitz. The methods avoid Bourgain-type spaces by exploiting the $\gamma\equiv1$ near the boundary and yield a practically relevant, constructive approach to boundary measurements for rough conductivities.
Abstract
We show the validity of Nachman's procedure (Ann. Math. 128(3):531-576, 1988) for reconstructing a conductivity $γ$ from its Dirichlet-to-Neumann map $Λ_γ$ for less regular conductivities, specifically $γ\in W^{3/2,2n}(Ω)$ such that $γ\equiv 1$ near $\partial Ω$. We also obtain a log-type stability estimate for the inverse problem when $γ$ has slightly higher regularity, i.e., $γ\in W^{2-s,n/s}(Ω)$ for $0 < s <1/2$.