TY - JOUR

T1 - On uniqueness of continuation for polynomials of solutions to second-order elliptic PDE

AU - Belishev, M. I.

AU - Filonov, N. D.

AU - Krymskiy, S. T.

AU - Vakulenko, A. F.

PY - 2020/1/4

Y1 - 2020/1/4

N2 - 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.

AB - 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.

KW - MANIFOLDS

KW - Second-order elliptic PDE

KW - polynomials of solutions

KW - uniqueness of continuation of polynomials

KW - uniqueness of continuation of solutions

UR - http://www.scopus.com/inward/record.url?scp=85078590628&partnerID=8YFLogxK

U2 - 10.1080/00036811.2019.1703959

DO - 10.1080/00036811.2019.1703959

M3 - Article

AN - SCOPUS:85078590628

JO - Applicable Analysis

JF - Applicable Analysis

SN - 0003-6811

ER -