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Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations. (English) Zbl 1202.35336
The authors consider a second-order elliptic operator \(A = - \partial_{x_0}^{2} - \nabla_{x}(c(x)\nabla_{x})\) in a bounded domain of \(\mathbb R^{n+1}\), \(n \geq 2\), where the coefficient \(c(x)\) is piecewise smooth yet discontinuous across a smooth interface \(S\). They prove a local Carleman estimate for \(A\) in the neighbourhood of any point of \(S\). The Calderón projector technique is used to obtain the estimate. Further, using the method of G. Lebeau and L. Robbiano [Commun. Partial Differ. Equations 20, No. 1–2, 335–356 (1995; Zbl 0819.35071)] they prove the null controllability for the linear parabolic initial problem with Dirichlet boundary conditions associated with the operator \(\partial_{t} - \nabla_{x}(c(x)\nabla_{x})\).
The results of this paper opens perspective for future research towards the null controllability of semi-linear parabolic equations in space of dimension \(n \geq 2\) and towards more complicated situations, for instance, in the case of coefficients with singularities that do not lie on a smooth interface.

MSC:
35R05 PDEs with low regular coefficients and/or low regular data
35J15 Second-order elliptic equations
35B45 A priori estimates in context of PDEs
93B05 Controllability
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