On the reconstruction of interfaces using complex geometrical optics solutions for the acoustic case

Abstract

We deal with the problem of reconstruction of the coefficient discontinuities (or supports) of scalar divergence form equations with lower order terms from the Dirichlet-to-Neumann map using complex geometrical optics (CGO) solutions. We consider both penetrable and impenetrable obstacles. The usual proofs for justifying this method assume, in addition to the smoothness of the coefficients and the interfaces, the following two conditions. The finiteness of the touching points of the phase’s level curves (or surfaces) of the used CGO solutions with the interface. The positivity of the lower bound of the relative curvature of the interface. In this paper, we show how we can remove these two conditions and justify the reconstruction method considering L∞ coefficients and Lipschitz interfaces of discontinuity.

Monotonicity and Enclosure Methods for the $p$-Laplace Equation

“…The enclosure method for linear equations allows one to identify various shapes by using CGO solutions with different phase functions. See for instance [22,28], which use linear phase functions to determine the convex hull, and [25] where spherical phase functions are used to reconstruct non-convex parts of the obstacle. The work [24] uses the enclosure method with CGO solutions with polynomial phases, where the energy is concentrated inside a cone, and in this case it is possible to approximate the exact shape of certain types of obstacles.…”

The Enclosure Method for the Anisotropic Maxwell System

“…This geometrical inverse problem is quite well studied in the literature see [4] and several methods have been proposed to solve it. In this paper, we focus on one of these method, called the enclosure method, which is initiated by Ikehata, see for examples [2,3], and developed by many researchers [7,9,14,18,19,20], [6,19] for the acoustic model, [5,9] for the Lamé model and [7,21] for the Maxwell model. The testing functions used in [7,21] are complex geometric optics (CGO) solutions of the isotropic Maxwell's equation.…”

Reconstruction of Interfaces from the Elastic Farfield Measurements Using CGO Solutions

“…The analysis is based on the use of integral equation methods on Sobolev spaces H s (∂D), s ∈ R, for the impenetrable case and L p estimates of the gradients of the solutions of the Lamé system with discontinuous Lamé coefficients for the penetrable case. This is a generalization to the Lamé system of the previous works [21] and [33] concerning the Maxwell and acoustic cases respectively. The paper is organized as follows.…”