Recovery of Non-Compactly Supported Coefficients of Elliptic Equations on an Infinite Waveguide

Abstract

We consider the unique recovery of a non compactly supported and non periodic perturbation of a Schrödinger operator in an unbounded cylindrical domain, also called waveguide, from boundary measurements. More precisely, we prove recovery of general class of electric potentials from the partial Dirichlet-to-Neumann map, where the Dirichlet data is supported on slightly more than half of the boundary and the Neumann data is taken on the other half of the boundary. We apply this result in different context including recovery of some general class of coefficients from measurements on a bounded subset and recovery of an electric potential, supported on an unbounded cylinder, of a Schrödinger operator in a slab.

Determination of non-compactly supported electromagnetic potentials in an unbounded closed waveguide

“…Here in contrast to [14,15,33,37] we do not restrict our analysis to compactly supported or periodic coefficients where, by mean of unique continuation or Floquet decomposition, one can transform the problem stated on an unbounded domain into a problem on a bounded domain. Like [30], we introduce a new class of CGO solutions designed for infinite cylindrical domains. The difficulties in the construction of such solutions are coming both from the fact that we consider magnetic potentials that are not compactly supported and the fact that we need to preserve the square integrability of the CGO solutions, which is not guarantied by the usual CGO solutions in unbounded domains.…”

“…Here the goal of the inverse problem can be described as the unique recovery of an electromagnetic impurity perturbing the guided propagation (see [10,25]). Let us also mention that in this paper we consider general closed waveguides, only subjected to condition (1.1), that have not necessary a cylindrical shape comparing to other related works like [14,15,30]. This means that we can consider our inverse problem in closed waveguides with different types of geometrical deformations, including bends and twisting, which can be used in several context for improving the propagation of signals (see for instance [46]).…”

“…In [14,15], the authors considered the stable recovery of coefficients periodic along the axis of an infinite cylindrical domain. More recently, [30] considered, for what seems to be the first time, the recovery of non-compactly supported and non-periodic electric potentials appearing in an infinite cylindrical domain. The results of [30] include also an extension of the work of [37] to the recovery of non-compactly supported coefficients in a slab.…”

“…More recently, [30] considered, for what seems to be the first time, the recovery of non-compactly supported and non-periodic electric potentials appearing in an infinite cylindrical domain. The results of [30] include also an extension of the work of [37] to the recovery of non-compactly supported coefficients in a slab. We mention also the work [1,2,16,26,28,31,32] treating the determination of coefficients appearing in different PDEs on an infinite cylindrical domain from boundary measurements.…”

“…Like in [18,43], we consider convexified weight, instead of the linear weight used in [30,Proposition 31], in order to be able to absorb first order perturbations of the Laplacian. Our first Carleman estimates can be seen as an extension of [18, Proposition 2.3], stated with linear weight, to unbounded cylindrical domains.…”