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
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 \textit{G. Lebeau} and \textit{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.
null-controllability of linear parabolic equations
scientific article; zbMATH DE number 5731005
scientific article