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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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References:
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