On two four-dimensional curl operators and their applications

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.