Standard

On uniqueness of continuation for polynomials of solutions to second-order elliptic PDE. / Belishev, M. I.; Filonov, N. D.; Krymskiy, S. T.; Vakulenko, A. F.

In: Applicable Analysis, 04.01.2020.

Research output: Contribution to journal › Article › peer-review

Harvard

Belishev, MI, Filonov, ND, Krymskiy, ST & Vakulenko, AF 2020, 'On uniqueness of continuation for polynomials of solutions to second-order elliptic PDE', Applicable Analysis. https://doi.org/10.1080/00036811.2019.1703959

APA

Belishev, M. I., Filonov, N. D., Krymskiy, S. T., & Vakulenko, A. F. (2020). On uniqueness of continuation for polynomials of solutions to second-order elliptic PDE. Applicable Analysis. https://doi.org/10.1080/00036811.2019.1703959

Vancouver

Belishev MI, Filonov ND, Krymskiy ST, Vakulenko AF. On uniqueness of continuation for polynomials of solutions to second-order elliptic PDE. Applicable Analysis. 2020 Jan 4. https://doi.org/10.1080/00036811.2019.1703959

Author

Belishev, M. I. ; Filonov, N. D. ; Krymskiy, S. T. ; Vakulenko, A. F. / On uniqueness of continuation for polynomials of solutions to second-order elliptic PDE. In: Applicable Analysis. 2020.

BibTeX

@article{e24cdd3dec064d1ebc2d40158732532c,
title = "On uniqueness of continuation for polynomials of solutions to second-order elliptic PDE",
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.",
keywords = "MANIFOLDS, Second-order elliptic PDE, polynomials of solutions, uniqueness of continuation of polynomials, uniqueness of continuation of solutions",
author = "Belishev, {M. I.} and Filonov, {N. D.} and Krymskiy, {S. T.} and Vakulenko, {A. F.}",
year = "2020",
month = jan,
day = "4",
doi = "10.1080/00036811.2019.1703959",
language = "English",
journal = "Applicable Analysis",
issn = "0003-6811",
publisher = "Taylor & Francis",

}

RIS

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 -

ID: 51315672