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Let $\Omega\subset\R^n$, $n\ge 3$, be a smooth bounded domain and consider acoupled 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 shownthat the coefficients $\gamma$, $\kappa$ and $A=(a_{jk})$ are uniquelydetermined from the knowledge of the boundary map $u|_{\partial\Omega}\mapsto\nu\cdot A\nabla \psi|_{\partial\Omega}$, where $\nu$ is the unit outer normalto $\partial\Omega$. The coupled system models the following physicalphenomenon. Given a fixed voltage distribution, maintained on the boundary$\partial\Omega$, an electric current distribution appears inside $\Omega$. Thecurrent in turn acts as a source of heat inside $\Omega$, and the heat flowsout of the body through the boundary. The boundary measurements above thencorrespond to the map taking a voltage distribution on the boundary to theresulting heat flow through the boundary. The presented mathematical resultssuggest a new hybrid diffuse imaging modality combining electrical prospectingand heat transfer-based probing.

Determining Electrical and Heat Transfer Parameters Using Coupled Boundary Measurements