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Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient, and applications. (English) Zbl 1182.35057
Summary: We study the observability and some of its consequences for the one-dimensional heat equation with a discontinuous coefficient (piecewise \({\mathcal C}^1\)). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields results of controllability to the trajectories for semilinear equations. It also yields a stability result for the inverse problem of the identification of the diffusion coefficient.

MSC:
35B45 A priori estimates in context of PDEs
35K05 Heat equation
35R05 PDEs with low regular coefficients and/or low regular data
93B07 Observability
35K58 Semilinear parabolic equations
35R30 Inverse problems for PDEs
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