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Carleman estimates for some non-smooth anisotropic media. (English) Zbl 1300.47052

The authors prove a Carleman estimate for the elliptic operator \(A = -\nabla\cdot(B\nabla)\) with an arbitrary observation region, where \(B\) is an \(n\times n\) block diagonal matrix \(B\) in which the first block \(C_\tau\) is a Hermitian matrix of order \((n - 1)\), the second block \(c\) is a positive function, and both are piecewise smooth in a bounded domain \(\Omega\subset {\mathbb R}^n\). Setting \(S\) for the set where discontinuities of \(C_\tau\) and \(c\) can occur, it is supposed that \(\Omega\) is stratified in a neighborhood of \(S\) in the sense that locally it takes the form \(\Omega'\times(-\delta,\delta)\) with \(\Omega'\subset {\mathbb R}^{n-1}\), \(\delta>0\) and \(S=\Omega'\times\{0\}\). This Carleman estimate is obtained through the introduction of a suitable mesh of the neighborhood of \(S\) and an associated approximation of \(c\) involving the Carleman large parameters.

MSC:

47F05 General theory of partial differential operators
35J15 Second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
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