Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

Patricia Gaitan (1, 2) , Hiroshi Isozaki (3) , Olivier Poisson (4) , Samuli Siltanen (5) , Janne Tamminen (6)

1 CPT - E8 Dynamique quantique et analyse spectrale
2 CPT - Centre de Physique Théorique - UMR 7332
3 Institute of Mathematics
4 LATP - Laboratoire d'Analyse, Topologie, Probabilités
5 Helsingin yliopisto = Helsingfors universitet = University of Helsinki
6 TUT - Tampere University of Technology [Tampere]

We consider an inverse boundary value problem for the heat equation ∂tu = div(γ∇xu) in (0,T)×Ω, u = f on (0,T)×∂Ω, u(t=0) = u_0, in a bounded domain Ω ⊂ Rn, n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time : γ(t,x) = k2 (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω \ D(t)). Fix a direction e∗ ∈ Sn−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 → ∂νu(t,x)(0,T)×∂Ω. The knowledge of the initial data u0 is not used in the proof. If we know min0≤t≤T supx∈D(t) {x· e∗}, we have the same conclusion from the local Dirichlet-to-Neumann map. The results have applications to nondestructive testing. Consider a physical body consisiting of homogeneous material with constant heat conductivity ex- cept for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux mea- surements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.

Domaines: Equations aux dérivées partielles [math.AP]

Patricia Gaitan, Hiroshi Isozaki, Olivier Poisson, Samuli Siltanen, Janne Tamminen. Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case. Communications in Partial Differential Equations, 2015, 40 (5), pp.837-877. ⟨10.1080/03605302.2014.992533⟩. ⟨hal-01255481⟩