Geodesic
Nonunique continuation for the Maxwell system
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 41, Tome 393 (2011), pp. 80-100

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We give an example of the stationary Maxwell system, which has nontrivial smooth solution with compact support; the coefficients $\varepsilon,\mu$ belong to $C^\alpha$ for all $\alpha1$. Our example shows that the stationary Maxwell system does not possess the unique continuation property in case of nonsmooth coefficients. 
@article{ZNSL_2011_393_a5,
     author = {M. N. Demchenko},
     title = {Nonunique continuation for the {Maxwell} system},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {80--100},
     publisher = {mathdoc},
     volume = {393},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_393_a5/}
}
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M. N. Demchenko. Nonunique continuation for the Maxwell system. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 41, Tome 393 (2011), pp. 80-100. http://geodesic.mathdoc.fr/item/ZNSL_2011_393_a5/