ON A SPECTRAL PROBLEM FOR THE LAPLACE OPERATOR WITH MORE GENERAL BOUNDARY CONDITIONS

In this paper, we consider a spectral problem for the Laplace operator with more general boundary conditions in a unit disk B₁. In the special cases, the boundary conditions inlude periodic and Samarskii-Ionkin type boundary conditions. The main important property of our problem is its non-self-adjointness, which causes number of difficulties in their analytical and numerical solutions. For example, the Fourier method of separation of variables cannot be applied directly to our problem. Therefore, the possibility of separation of variables is justified in this paper. Namely, we present a method that reduces solution of the problem to a sequential solution of two classical local boundary value problems. By using this method, we construct all eigenfunctions and eigenvalues of the problem in explicit forms. Moreover, completeness of the system of the eigenfunctions is proved in L² (B₁). Notably, our result generalises the special case of the result on the two-dimensional periodic boundary value problem for the Laplace operator obtained in [1–2].

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