Determining Electrical and Heat Transfer Parameters Using Coupled Boundary Measurements

Let Ω ⊂ R, n ≥ 3, be a smooth bounded domain and consider a coupled system in Ω consisting of a conductivity equation ∇ · γ(x)∇u(t, x) = 0 and an anisotropic heat equation κ(x)∂tψ(t, x) = ∇ · (A(x)∇ψ(t, x)) + (γ∇u(t, x)) · ∇u(t, x), t ≥ 0. It is shown that the coefficients γ, κ and A = (ajk) are uniquely determined from the knowledge of the boundary map u|∂Ω 7→ ν ·A∇ψ|∂Ω, where ν is the unit outer normal to ∂Ω. The coupled system models the following physical phenomenon. Given a fixed voltage distribution, maintained on the boundary ∂Ω, 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.