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Articles

On uniqueness of continuation for polynomials of solutions to second-order elliptic PDE

, , &
Pages 3689-3696 | Received 23 Sep 2019, Accepted 05 Dec 2019, Published online: 04 Jan 2020
 

Abstract

Let L=i,jaijxixj+ibixi+c be an elliptic operator with smooth enough coefficients, u a solution to the equation Lu = 0 in ΩRn, ωΩ an open set. As is well known, if u|ω=0 then u = 0 everywhere in Ω. Let P be a polynomial of variables τ1,,τm, functions u1,,um the solutions to Lu = 0; let p(x)=P(u1(x),,um(x)). If the coefficients of the operator are (real) analytic then u1,,um,p are analytic and p|ω=0 implies p0 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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Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the RFBR grant 18-01-00269 and Volks-Wagen Foundation.

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