Carmen Perugia

A gap in the essential spectrum of a cylindrical waveguide with a periodic aperturbation of the surface

It is proved that small periodic singular perturbation of a cylindrical waveguide surface may open a gap in the continuous spectrum of the Dirichlet problem for the Laplace operator. If the perturbation period is long and the caverns in the cylinder are small, the gap certainly opens.

Uniform resolvent convergence for strip with fast oscillating boundary

Abstract In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary we impose Dirichlet, Neumann or Robin boundary condition. In all cases we describe the homogenized operator, establish the uniform resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. These results are obtained as the order of the amplitude of the oscillations is less, equal or greater than that of the period. It is shown that under the homogenization the type of the boundary condition can change.

Optimal control for a second-order linear evolution problem in a domain with oscillating boundary

This paper is concerned with the study of homogenization of an optimal control problem governed by a second-order linear evolution equation with a homogeneous Neumann boundary condition in a domain bounded at the bottom by a smooth wall and at the top by a rough wall. The latter is assumed to consist in a plane wall covered with periodically distributed asperities, with a fixed height, whose size depends on a small parameter . We identify the limit problem and we remark that both limit state equation and limit cost are different from those ones at level.

Estimates in homogenization of degenerate elliptic equations by spectral method

We study the homogenization of elliptic equations stated in L 2 -space with degenerate weight. Both coefficients of the differential operator and the weight are e-periodic and highly oscillating as e tends to zero. Under minimal hypotheses on the coefficients and the weight we prove estimates of order e and e 2 for L 2 -norm of the difference between the exact solution and its appropriate approximations by L 2 -norm of the right-side function. The spectral method based on Bloch decomposition is used. In the case of nonunique solution provided that the weight is not regular we consider estimates for any of so-called