Determining electrical and heat transfer parameters using coupled boundary measurements

Abstract

Let $\Omega \subset {\mathbb{R}}^n$, $n\ge 3$, be a smooth bounded domain and consider a coupled system in $\Omega$ consisting of a conductivity equation $\nabla \cdot \gamma (x)\nabla u(t,x)=0$ and an anisotropic heat equation ${\kappa }^{-1}(x){\partial }_t\psi (t,x)=\nabla \cdot (A(x)\nabla \psi (t,x))+(\gamma \nabla u(t,x))\cdot \nabla u(t,x),t\ge 0$. It is shown that the coefficients $\gamma$, $\kappa$ and $A=(a_{jk})$ are uniquely determined from the knowledge of the boundary map $u{|}_{\partial \Omega }\mapsto \nu \cdot A\nabla \psi {|}_{\partial \Omega }$, where $\nu$ is the unit outer normal to $\partial \Omega$. The coupled system models the following physical phenomenon. Given a fixed voltage distribution, maintained on the boundary $\partial \Omega$, an electric current distribution appears inside $\Omega$. The current in turn acts as a source of heat inside $\Omega$, and the heat flows out of the body through the boundary. The boundary measurements above then correspond to the map taking a voltage distribution on the boundary to the resulting heat flow through the boundary. The presented mathematical results suggest a new hybrid diffuse imaging modality combining electrical prospecting and heat transfer-based probing.