We consider a linear parabolic transmission problem across an interface of codimension one in a bounded domain or on a Riemannian manifold, where the transmission conditions involve an additional parabolic operator on the interface. This system is an idealization of a three-layer model in which the central layer has a small thickness . We prove a Carleman estimate in the \nhd of the interface for an associated elliptic operator by means of partial estimates in several microlocal regions. In turn, from the Carleman estimate, we obtain a spectral inequality that yields the null-controllability of the parabolic system. These results are uniform with respect to the small parameter .
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Jérôme Le Rousseau, Matthieu Léautaud, Luc Robbiano, Controllability of a parabolic system with a diffuse interface. J. Eur. Math. Soc. 15 (2013), no. 4, pp. 1485–1574DOI 10.4171/JEMS/397