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Smooth counterexamples to strong unique continuation for a Beltrami system in $$\mathbb C^2$$. (English) Zbl 1261.35032
Summary: We construct an example of a smooth map $$\mathbb C\to\mathbb C^2$$ which vanishes to infinite order at the origin, and such that the ratio of the norm of the $$\overline z$$ derivative to the norm of the $$z$$ derivative also vanishes to infinite order. This gives a counterexample to strong unique continuation for a vector valued analogue of the Beltrami equation.

##### MSC:
 35B60 Continuation and prolongation of solutions to PDEs 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 30G20 Generalizations of Bers and Vekua type (pseudoanalytic, $$p$$-analytic, etc.) 32W50 Other partial differential equations of complex analysis in several variables 35J46 First-order elliptic systems
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