International audienceThis work deals with an inverse boundary value problem arising from the equation of heat conduction. Mathematical theory and algorithm is described in dimensions 1–3 for probing the discontinuous part of the conductivity from local temperature and heat flow measurements at the boundary. The approach is based on the use of complex spherical waves, and no knowledge is needed about the initial temperature distribution. In dimension two we show how conformal transformations can be used for probing deeper than is possible with discs. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method is capable of recovering locations of discontinuities approximately from noisy data
We consider the heat equation ∂ t y − div(c∇y) = H with a discontinuous coefficient in three connected situations. We give uniqueness and stability results for the diffusion coefficient c(•) in the main case from measurements of the solution on an arbitrary part of the boundary and at a fixed time in the whole spatial domain. The diffusion coefficient is assumed to be discontinuous across an unknown interface. The key ingredients are a Carleman-type estimate with non-smooth data near the interface and a stability result for the discontinuous coefficient c(•) in an inverse problem associated with the stationary equation −div(c∇u) = f .
Abstract. We consider the heat equation with a diffusion coefficient that is discontinuous at an interface. We give global Carleman estimates for solutions of this equation, even if the jump of the coefficient across the interface has not a constant sign.
International audienceWe consider an inverse boundary value problem for the heat equation ∂tv = divx (γ∇xv) in (0, T) × Ω, where Ω is a bounded domain of R 3 , the heat conductivity γ(t, x) admits a surface of discontinuity which depends on time and without any spatial smoothness. The reconstruction and, implicitly, uniqueness of the moving inclusion, from the knowledge of the Dirichlet-to-Neumann operator, is realised by a dynamical probe method based on the construction of fundamental solutions of the elliptic operator −∆ + τ 2 ·, where τ is a large real parameter, and a couple of inequalities relating data and integrals on the inclusion, which are similar to the elliptic case. That these solutions depend not only on the pole of the fundamental solution, but on the large parameter τ also, allows the method to work in the very general situation
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