On eigenfunctions and eigenvalues of some boundary value problems for a nonlocal biharmonic operator
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Keywords:
Spectral problem, nonlocal operator, biharmonic operator, Dirichlet problem, Neumann problem, Samarsky-Ionkin problem, eigenfunctions, eigenvalues, attached functions, completeness.Abstract
In this note, the concept of a nonlocal biharmonic operator is introduced. When
introducing this operator, mappings of the type of involution are used. Namely, in the differential
expression of this operator, in addition to the variables x (x1, x2 ,..., xn ) , transformed arguments
with mappings of the form 1 1 1 ,..., , , ,..., ,1 j j j j j n S x x x p x x x j n and their multiplication
also involved. Spectral problems with Dirichlet and Neumann-type boundary conditions are
considered in an n-dimensional parallelepiped for a given nonlocal biharmonic operator. The
eigenfunctions and eigenvalues of the problems under consideration are explicitly constructed.
When constructing these elements, eigenfunctions and eigenvalues of the classical biharmonic
operator with Dirichlet and Neumann type boundary conditions are essentially used. Theorems on
the orthonomization and completeness of the systems of eigenfunctions of the problems under
consideration are proved. Examples of the corresponding parameters for special cases involved in
the problems under consideration are given. In addition, in the two-dimensional case for the
corresponding nonlocal biharmonic operator, spectral issues of boundary value problems of the
Samarsky-Ionkin type are also investigated. The proper and attached functions of the problem under
consideration are found and theorems on the completeness of these systems are proved.
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