Abstract
Let be an elliptic operator with smooth enough coefficients, u a solution to the equation Lu = 0 in , an open set. As is well known, if then u = 0 everywhere in Ω. Let P be a polynomial of variables , functions the solutions to Lu = 0; let . If the coefficients of the operator are (real) analytic then are analytic and implies in Ω. Is the same true for smooth but not analytic coefficients? The question also concerns the polynomials of harmonic quaternion fields. In general, the answer turns out to be negative: the paper provides the relevant counterexamples.
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