On the strong unique continuation principle for inequalities of Maxwell type (Q915958)

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 \(\alpha\).