Global Uniqueness for the Calderón Problem With Lipschitz Conductivities

Abstract

We prove uniqueness for the Calderón problem with Lipschitz conductivities in higher dimensions. Combined with the recent work of Haberman, who treated the three-and four-dimensional cases, this confirms a conjecture of Uhlmann. Our proof builds on the work of Sylvester and Uhlmann, Brown, and Haberman and Tataru who proved uniqueness for C 1 -conductivities and Lipschitz conductivities sufficiently close to the identity.

Anomaly Detection in Random Heterogeneous Media

“…In two dimensions and for isotropic κ ∈ L ∞ + (D), the question of uniqueness was answered affirmatively in the celebrated paper [7] by Astala and Päivärinta. The recent result by Caro and Rogers [30] yields the same assertion for d ≥ 3, provided that κ is a Lipschitz function. When the conductivity is not assumed to be isotropic, there is always an obstruction to uniqueness, namely we have Λ κ1 = Λ κ2 , whenever κ 2 = F * κ 1 is the pushforward conductivity by a diffeomorphism F on D that leaves the boundary ∂D invariant.…”

Sufficient Conditions for the Existence of Limiting Carleman Weights

“…Let W be a Weyl operator with an eigenflag direction L = v . The operator W | v∧v ⊥ is symmetric, and diagonalizes in an orthonormal basis v ∧e 2 , v ∧e 3 , v ∧e 4 . Define e 1 = v and e i j = e i ∧e j .…”

“…https://doi.org/10.1017/fms.2017. 4 The consequence of this is that λ 1 = λ 2 = 0, and every Lie bracket between elements in the basis {A, B, ∂ y , ∂ z } vanishes, and hence they form a coordinate basis for some set of coordinates about p. We keep on denoting them (t, x, y, z), although only the last two would coincide with the former. The first two, (t, x), will still, however, parametrize the first factor of the product structure of M. In this chart,g is written as…”

“…Since the 1987 foundational paper of Sylvester and Uhlmann [18] (for more recent results see [4,10,11]), the only effective strategy to solve the Calderón inverse problem has been based on the existence of Complex Geometric Optic solutions. In the Riemannian setting, it was discovered in [8] that this type of solution depends on the existence of so called limiting Carleman weights.…”

The inverse conductivity problem via the calculus of functions of bounded variation

“…Uniqueness results for less smooth conductivities in dimensions three and higher were obtained in Brown and Torres 9 and Krupchyk and Uhlmann. 10 Further, works concerning uniqueness in 3D can be found in Haberman and Tataru 11 and Caro and Rogers 12 combined with the recent work of Haberman 13 for W 1,n (Ω) (n = 3, 4) conductivities.…”