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JOURNALS // Sibirskii Matematicheskii Zhurnal // Archive

Sibirsk. Mat. Zh., 2022 Volume 63, Number 6, Pages 1266–1275 (Mi smj7730)

The uniqueness criterion for a solution to a boundary value problem for the operator $\frac{\partial ^{2p} }{\partial t^{2p}}-A$ with an elliptic operator $A$ of arbitrary order

B. E. Kanguzhinab, B. D. Koshanovca

a Al-Farabi Kazakh National University
b Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science, Republic of Kazakhstan, Almaty
c International Information Technology University

Abstract: We establish the uniqueness criterion for a solution to the operator $\frac{\partial ^{2p}}{\partial t^{2p}}-A(x,D)$ with the Dirichlet time-dependent boundary conditions and general boundary conditions in the space variables. The order of the differentiation operator $\frac{\partial ^{2p} }{\partial t^{2p}}$ is assumed even. Note that $A(x,D)$ in the space variables is an arbitrary elliptic operator with some rather general boundary operators $B_j$ obeying the conventional Agmon conditions. The Agmon conditions ensure the existence of a complete orthonormal system of eigenfunctions (in $L_2(\Omega)$) provided that $\Omega$ is a bounded domain with sufficiently smooth boundary.

Keywords: higher order elliptic operator, boundary value problem, eigenfunction, uniqueness of a solution, entire function of exponential type.

UDC: 517.956

MSC: 35R30

Received: 12.01.2022
Revised: 25.04.2022
Accepted: 15.06.2022

DOI: 10.33048/smzh.2022.63.608


 English version:
Siberian Mathematical Journal, 2022, 63:6, 1083–1090

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© Steklov Math. Inst. of RAS, 2025