On two four-dimensional curl operators and their applications
Jenaliyev M.T. Kassymbekova A.S. Yergaliyev M.G.
2025E.A. Buketov Karaganda University Publish house
Bulletin of the Karaganda University. Mathematics Series
2025#118Issue 2106 - 121 pp.
Academician O.A. Ladyzhenskaya emphasized the importance of constructing a fundamental system in the space of solenoidal functions for simple domains such as squares, cubes, and similar regions. This article examines the problem of constructing such fundamental systems for a four-dimensional parallelepiped and cube. As is well known, applying the stream functions known from the two- and three-dimensional cases, the spectral problem for the Stokes operator reduces to the so-called clamped plate problem, which, in turn, has no solution in domains such as the square, cube, or parallelepiped. Thus, in higher-dimensional cases, the necessity of an analogous stream function becomes evident. In this work, the authors propose two curl operators that satisfy the above-mentioned requirements. Using the introduced curl operators, the spectral problem for the biharmonic operator in a four-dimensional parallelepiped and cube is formulated. Alternative approaches to constructing a fundamental system are presented, given the unsolvability of the spectral problem. Furthermore, the growth orders of the obtained eigenvalues are established.
curl operator , fundamental system , spectral problem
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Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan
Institute of Mathematics and Mathematical Modeling
Institute of Mathematics and Mathematical Modeling
Institute of Mathematics and Mathematical Modeling
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