The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary
Vladimir Arkad'evich Kozlov, С. А. Назаров · St Petersburg Mathematical Journal · 2011
Asymptotic expansions are constructed for the eigenvalues of the Dirichlet problem for the biharmonic operator in a domain with highly indented and rapidly oscillating boundary (the Kirchhoff model of a thin plate). The asymptotic constructions depend heavily on the quantity $\gamma$ that describes the depth $O(\varepsilon ^\gamma )$ of irregularity ($\varepsilon$ is the oscillation period). The resulting formulas relate the eigenvalues in domains with close irregular boundaries and make it possible, in particular, to control the order of perturbation and to find conditions ensuring the validity (or violation) of the classical Hadamard formula.