#### Vol. 6, No. 4, 2013

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Cauchy problem for ultrasound-modulated EIT

### Guillaume Bal

Vol. 6 (2013), No. 4, 751–775
##### 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 $\sigma \left(x\right)$ for $x\in X$, a bounded domain in ${ℝ}^{n}$, from knowledge of $\sigma \left(x\right)|\nabla u{|}^{2}\left(x\right)$ for $x\in X$, where $u$ is the solution to the elliptic equation $-\nabla \cdot \sigma \nabla u=0$ 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 $\sigma \left(x\right)$ 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