7 research outputs found
Spectral analysis of the biharmonic operator subject to Neumann boundary conditions on dumbbell domains
We consider the biharmonic operator subject to homogeneous boundary
conditions of Neumann type on a planar dumbbell domain which consists of two
disjoint domains connected by a thin channel. We analyse the spectral behaviour
of the operator, characterizing the limit of the eigenvalues and of the
eigenprojections as the thickness of the channel goes to zero. In applications
to linear elasticity, the fourth order operator under consideration is related
to the deformation of a free elastic plate, a part of which shrinks to a
segment. In contrast to what happens with the classical second order case, it
turns out that the limiting equation is here distorted by a strange factor
depending on a parameter which plays the role of the Poisson coefficient of the
represented plate.Comment: To appear in "Integral Equations and Operator Theory
Estimates of Initial Scales for Layers with Small Random Negative-Definite Perturbations
Asymptotics of the frequency of a surface wave trapped by a slightly inclined barrier in a liquid layer
A two-dimensional statement of the scattering problem for an oblique incident surface wave by an obstacle in the form of a submerged barrier is considered. If the barrier is vertical, the discrete spectrum of the problem is shown to be empty, but for an inclined barrier an eigenvalue appears below the threshold of the continuous spectrum and the corresponding trapped mode decays exponentially in the direction perpendicular to the obstacle. The behaviour of the eigenvalue is analyzed for small values of the angle of inclination from the vertica