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

M. I. Belishev, N. D. Filonov, S. T. Krymskiy, A. F. Vakulenko

Research output: Contribution to journalArticlepeer-review

Abstract

Let L= Sigma(i,j) a(ij)partial derivative x(i)partial derivative(x)j + Sigma(i)b(i)partial derivative x(i) + c be an elliptic operator with smooth enough coefficients, u a solution to the equation Lu = 0 in Omega subset of R-n, omega subset of Omega an open set. As is well known, if then u vertical bar(omega) = 0 then u = 0 everywhere in Omega. Let P be a polynomial of variables tau(1), ..., tau(m), functions u(1), ..., u(m) the solutions to Lu = 0; let p(x) = P(u(1)(x), ...., u(m)(x)). If the coefficients of the operator are (real) u(1), ..., u(m),p analytic then p vertical bar(omega) = 0 implies p = 0 in Omega. 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.

Original languageEnglish
Number of pages8
JournalApplicable Analysis
Early online date4 Jan 2020
DOIs
StateE-pub ahead of print - 4 Jan 2020

Keywords

  • MANIFOLDS
  • Second-order elliptic PDE
  • polynomials of solutions
  • uniqueness of continuation of polynomials
  • uniqueness of continuation of solutions

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