Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case |
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Authors: | P Gaitan H Isozaki O Poisson S Siltanen |
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Institution: | 1. Centre de Physique Théorique , Aix Marseille Université , Aix en Provance , France;2. Institute of Mathematics , University of Tsukuba , Tsukuba , Japan;3. Institut de Mathématiques de Marseille , Aix Marseille Université , Aix en Provance , France;4. Department of Mathematics and Statistics , University of Helsinki , Finland |
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Abstract: | We consider an inverse boundary value problem for the heat equation ? t u = div (γ? x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| t=0 = u 0, in a bounded domain Ω ? ? n , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k 2 (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?ν u(t, x)|(0, T)×?Ω. The knowledge of the initial data u 0 is not used in the proof. If we know min0≤t≤T (sup x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking. |
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Keywords: | Dirichlet-to-Neumann map Heat probing Inverse problem Parabolic |
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