Cauchy problem for ultrasound-modulated EIT

Bal, Guillaume

doi:10.2140/apde.2013.6.751

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Cauchy problem for ultrasound-modulated EIT

Guillaume Bal

Vol. 6 (2013), No. 4, 751–775

DOI: 10.2140/apde.2013.6.751

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 $σ (x)$ for $x \in X$ , a bounded domain in $ℝ^{n}$ , from knowledge of $σ (x) | \nabla u |^{2} (x)$ for $x \in X$ , where $u$ is the solution to the elliptic equation $- \nabla \cdot σ \nabla u = 0$ in $X$ with $u = f$ on $\partial X$ .

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, $p = 1$ 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 $0$ -Laplacian. In general, the diffusion coefficient $σ (x)$ can be stably reconstructed only on a subset of $X$ described as the domain of influence of the space-like part of the boundary $\partial X$ for an appropriate Lorentzian metric. Global reconstructions for specific geometries or based on the construction of appropriate complex geometric optics solutions are also analyzed.

Keywords

hybrid inverse problems, nonlinear hyperbolic equations, power density internal functional

Mathematical Subject Classification 2010

Primary: 35B38, 35R30, 35CXX, 35L72

Milestones

Received: 3 April 2011

Revised: 18 December 2011

Accepted: 26 June 2012

Published: 21 August 2013

Authors

Guillaume Bal
Deparment of Applied Physics & Applied Mathematics Columbia University 500 W. 120th Street S.W. Mudd Building Room 206 New York, NY 10027 United States

Vol. 6, No. 4, 2013