On the strong unique continuation principle for inequalities of Maxwell type. (English) Zbl 0703.35028

For inequalities of Maxwell type \[ | curl a| +| div(\alpha a)| \leq cr^{\epsilon -1}| a| \] there is proved the strong unique continuation principle; i.e. a vanishes identically in a neighborhood of zero if a decays of infinite order at zero. The definite matrix valued function \(\alpha\) (x) satisfies the hypothesis \[ | \partial_ j\alpha (x)| \leq c| x|^{\epsilon -1}\quad (j=1,2,3) \] for some \(\epsilon >0\). In the case of a real function \(\alpha\) (x) the weaker condition \(| \partial_ r\alpha (x)| \leq c| x|^{\epsilon -1}\) is sufficient.
The proof relies on Carleman inequalities in \(L^ 2\) and suitable representations of curl and divergence in polar coordinates including especially the perturbation \(\alpha\).
Reviewer: V.Vogelsang


35B60 Continuation and prolongation of solutions to PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
35R45 Partial differential inequalities and systems of partial differential inequalities
78A25 Electromagnetic theory (general)
Full Text: DOI EuDML


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