Determining Electrical and Heat Transfer Parameters Using Coupled Boundary Measurements

Let $\Omega\subset\R^n$, $n\ge 3$, 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 $\kappa^{-1}(x)\partial_t\psi(t,x)=\nabla\cdot (A(x)\nabla \psi(t,x))+(\gamma\nabla u(t,x))\cdot \nabla u(t,x), \quad t\ge 0$. 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.