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Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems. (English) Zbl 1038.35159

Cioranescu, Doina (ed.) et al., Nonlinear partial differential equations and their applications. Collège de France seminar. Vol. XIV. Lectures held at the J. L. Lions seminar on applied mathematics, Paris, France, 1997–1998. Amsterdam: Elsevier (ISBN 0-444-51103-2/hbk). Stud. Math. Appl. 31, 329-349 (2002).
As the authors emphasize: “The unique continuation problem is fairly well understood for scalar equations, for systems it is considerably less explored.”
In order to prove unique continuation results for systems the authors exploit the classical Carleman estimate of Hörmander and the more recent Carleman estimate for a scalar operator with partial analytic coefficients in the principal part. More precisely the authors consider the following type of systems \[ P_{j}u_{j}+b_{j} ( x,t;\nabla u) + c_{j} (x,t;u) =f_{j}, \quad j=1,\dots,m \tag{1} \] where \(P_{j}\) are second order operators with real principal part and \(C^{1}\) (or analytic in \(t\)) coefficients and other coefficients in \(L_{\text{loc}}^{\infty }\), \(b_{j}\), \(c_{j}\) are linear functions of \(\nabla u\) and \(u\) with \( L_{\text{loc}}^{\infty }\) coefficients, \(u=( u_{1}, \dots ,u_{m}) .\) For such systems they prove some Carleman estimates. In turn from the proved Carleman estimates the authors find some uniqueness and stability results for the Cauchy problem for the above type of systems.
In a subsequent part of the paper it is proven that the Maxwell system and the time dependent classical elasticity system can be reduced, by simple calculations, to system (1), so the uniqueness and stability results proven follow from the general case.
For the entire collection see [Zbl 0992.00032].

MSC:

35R30 Inverse problems for PDEs
35Q72 Other PDE from mechanics (MSC2000)
35Q60 PDEs in connection with optics and electromagnetic theory
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