Spectral gaps in the Dirichlet problem for the biharmonic operator on a plane periodically perforated by circular holes

摘要

The spectra of various operators in periodic media have a zone structure, which implies that spectral gaps, i.e., intervals on the real positive semiaxis that don't include the spectrum (though the ends of the intervals belong to the spectrum), may appear. The spectrum of the Dirichlet problem for the biharmonic operator on a plane perforated by a double periodic family of circular holes is investigated in this paper. When the radiuses of these holes reach certain values, the plane is reduced to a countable union of bounded sets. The spectrum of the indicated problem is, to an extent, discrete in this extreme case, so there is a possibility that the spectrum of the problem close to the limit one has arbitrarily many gaps. This statement is the one proved in this paper. It is shown that, if two eigenvalues of the limit problem are different, then the corresponding bands of the continuous spectrum of the problem close to the limit one have no common points. When the eigenvalues of the limit problem coincide, the mentioned methods are unable to detect the formation of a gap between the corresponding bands of the continuous spectrum. To validate the formation of gaps, the positions of the eigenvalues of the model problem are localized on the periodicity cell. The maximinimal approach and the specific weight estimates for the eigenfunctions are used in this case.