Unique localization of unknown boundaries in a conducting medium from boundary measurements

Canuto, Bruno

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 1-22

Résumé

We consider the problem of localizing an inaccessible piece $I$ of the boundary of a conducting medium $\Omega$, and a cavity $D$ contained in $\Omega$, from boundary measurements on the accessible part $A$ of $\partial \Omega$. Assuming that $g(t,\sigma )$ is the given thermal flux for $\left(t,\sigma \right)\in (0,T)\times A$, and that the corresponding output datum is the temperature $u(T_0,\sigma )$ measured at a given time $T_0$ for $\sigma \in A_{\mathrm{out}}\subset A$, we prove that $I$ and $D$ are uniquely localized from knowledge of all possible pairs of input-output data $(g,u{\left(T_0\right)}_{\mid A_{\mathrm{out}}})$. The same result holds when a mean value of the temperature is measured over a small interval of time.

Détail

DOI : 10.1051/cocv:2002001

Classification : 35R30
Keywords: inverse boundary value problems, cavities, corrosion, uniqueness