Determining nonsmooth first order terms from partial boundary measurements

Abstract

Abstract. We extend results of Dos Santos Ferreira-Kenig-Sjöstrand-Uhlmann (arXiv:math.AP/0601466) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schrödinger operator determine uniquely the magnetic field related to a Hölder continuous potential. We give a similar result for determining a convection term. The proofs involve Carleman estimates, a smoothing procedure, and an extension of the Nakamura-Uhlmann pseudodifferential conjugation method to logarithmic Carleman weights.

Inverse Problems With Partial Data for a Magnetic Schrödinger Operator in an Infinite Slab and on a Bounded Domain

“…This is due, in particular, to the fact that a reflection argument with respect to a boundary hyperplane leads to a magnetic potential which is in general only Lipschitz continuous. The construction of complex geometric optics solutions in this case is consequently more complicated, as already seen in [17] and [35].…”

“…In the presence of a magnetic potential, the inverse problem of determining the magnetic field and the electric potential from partial boundary measurements was addressed in [17], when γ 1 = ∂Ω and γ 2 is possibly a very small subset of the boundary, see also [35]. Under the assumption that A (1) = A (2) and q (1) = q (2) in a neighborhood of the boundary, in [7] it is proven that the magnetic field and the electric potential can be uniquely determined by boundary measurements, provided that γ 1 = ∂Ω and γ 2 is arbitrary.…”

Inverse problems with partial data for a Dirac system: A Carleman estimate approach

“…The last expression is related to a (nonlinear) two-plane transform of ∇ × (A 1 − A 2 ) over a set of two-planes. We may now apply the arguments in [7, Section 5] (see also [16,Section 7], where the last identity is the same as formula (40)). One first shows by complex analytic methods that the identity remains true with e i(φ 1 −φ 2 ) and g replaced by 1.…”

Complex Geometrical Optics and Calderón’s Problem

“…The CGO solutions constructed in this section have been applied to solve inverse boundary problems for the magnetic Schrödinger equation with different regularity in the coefficients [30], [35], [36], [42], [19], [26]. Also applications to an inverse boundary problem for linear elasticity are given in [31] (see also [7]).…”