We study the discrete spectra of boundary value problems for the biharmonic operator describing oscillations of a Kirchhoff plate in the form of a locally perturbed strip with rigidly clamped or simply supported edges. The two methods are applied: variational and asymptotic. The first method shows that for a narrowing plate the discrete spectrum is empty in both cases, whereas for a widening plate at least one eigenvalue appears below the continuous spectrum cutoff for rigidly clamped edges. The presence of the discrete spectrum remains an open question for simply supported edges. The asymptotic method works only for small variations of the boundary. While for a small smooth perturbation the construction of asymptotics is generally the same for both types of boundary conditions, the asymptotic formulas for eigenvalues can differ substantially even in the main correction term for a perturbation with corner points.
Original languageRussian
Pages (from-to)297-313
JournalСИБИРСКИЙ МАТЕМАТИЧЕСКИЙ ЖУРНАЛ
Volume61
Issue number2
StatePublished - 2020

    Research areas

  • Biharmonic operator, discrete spectrum, eigenvalue asymptotics, infinite Kirchhoff plate, асимптотика собственных чисел, бесконечная пластина Кирхгофа, бигармонический оператор, дискретный спектр

ID: 78546581