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.
- Second-order elliptic PDE
- polynomials of solutions
- uniqueness of continuation of polynomials
- uniqueness of continuation of solutions