Cauchy problem for ultrasound-modulated EIT

Abstract

Ultrasound modulation of electrical or optical properties of materials offers the possibility of devising hybrid imaging techniques that combine the high electrical or optical contrast observed in many settings of interest with the high resolution of ultrasound. Mathematically, these modalities require that we reconstruct a diffusion coefficient. x/ for x2 X, a bounded domain in R n , from knowledge of. x/jruj 2 .x/ for x2 X , where u is the solution to the elliptic equationr ruD 0 in X with uD f on@ X . This inverse problem may be recast as a nonlinear equation, which formally takes the form of a 0-Laplacian. Whereas p-Laplacians with p> 1 are well-studied variational elliptic nonlinear equations, pD 1 is a limiting case with a convex but not strictly convex functional, and the case p < 1 admits a variational formulation with a functional that is not convex. In this paper, we augment the equation for the 0-Laplacian with Cauchy data at the domain’s boundary, which results in a formally overdetermined, nonlinear hyperbolic equation. This paper presents existence, uniqueness, and stability results for the Cauchy problem of the 0Laplacian. In general, the diffusion coefficient . x/ can be stably reconstructed only on a subset of X described as the domain of influence of the space-like part of the boundary @ X for an appropriate Lorentzian metric. Global reconstructions for specific geometries or based on the construction of appropriate complex geometric optics solutions are also analyzed.