A bilinear strategy for Calderón's problem

Abstract

Electrical Impedance Imaging would suffer a serious obstruction if two different conductivities yielded the same measurements of potential and current at the boundary. The Calderón's problem is to decide whether the conductivity is indeed uniquely determined by the data at the boundary. In R d , for d ě 5, we show that uniqueness holds when the conductivity is in W 1`d´5 2p`, p pΩq, for d ď p ă 8. This improves on recent results of Haberman, and of Ham, Kwon and Lee. The main novelty of the proof is an extension of Tao's Bilinear Theorem.

Recent progress in the conductivity reconstruction in Calderón’s problem

“…Bilinear estimate [39] The observation of the table makes us wonder how much it would be interesting to check whether it is possible to prove Brown's conjecture [11], which affirms that in three and higher dimensions γ ∈ W 1,n is the minimum possible regularity for which uniqueness holds. Notice that the approaches used in [22,24,36] are not useful for reconstructing γ, because the proofs there are not constructive, meaning that they did not give a procedure to recover γ from Λ γ .…”

Inverse boundary value problems for polyharmonic operators with non-smooth coefficients

“…Further extensions include work of Caro and Rogers [14] who establish uniqueness for Lipschitz conductivities and an additional work of Haberman [18] that considers conductivities with one derivative in L d (where d is the dimension) at least for d = 3, 4. Recent extensions of these ideas may be found in articles by Ham, Kwon and Lee [20] and Ponce-Vanegas [27,28]. It is a conjecture of Uhlmann that the least regular Sobolev space for which we can prove uniqueness is W 1,d .…”