Determining electrical and heat transfer parameters using coupled boundary measurements

Let $Ω\subset\R^n$, $n\ge 3$, be a smooth bounded domain and consider a coupled system in $Ω$ consisting of a conductivity equation $\nabla \cdot γ(x) \nabla u(t,x)=0$ and an anisotropic heat equation $κ^{-1}(x)\partial_tψ(t,x)=\nabla\cdot (A(x)\nabla ψ(t,x))+(γ\nabla u(t,x))\cdot \nabla u(t,x), \quad t\ge 0$. It is shown that the coefficients $γ$, $κ$ and $A=(a_{jk})$ are uniquely determined from the knowledge of the boundary map $u|_{\partialΩ}\mapsto ν\cdot A\nabla ψ|_{\partialΩ}$, where $ν$ is the unit outer normal to $\partialΩ$. The coupled system models the following physical phenomenon. Given a fixed voltage distribution, maintained on the boundary $\partialΩ$, an electric current distribution appears inside $Ω$. The current in turn acts as a source of heat inside $Ω$, 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.

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