Giuseppe Cardone

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.

The Localization Effect for Eigenfunctions of the Mixed Boundary Value Problem in a Thin Cylinder with Distorted Ends

A simple sufficient condition on a curved end of a straight cylinder is found that provides a localization of the principal eigenfunction of the mixed boundary value for the Laplace operator with the Dirichlet conditions on the lateral side. Namely, when the small parameter, i.e., the ratio between the diameter and the length of the cylinder, tends to zero, the eigenfunction concentrates in the vicinity of the ends and decays exponentially in the interior. Similar effects are observed in the Dirichlet and Neumann problems, too.

On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition

We consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resolvent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results are the uniform resolvent convergence for the operator on the cell of periodicity obtained by the Floquet–Bloch decomposition, the two terms asymptotics for the band functions, and the complete asymptotic expansion for the bottom of the spectrum with an exponentially small error term.

Asymptotic Analysis, Polarization Matrices, and Topological Derivatives for Piezoelectric Materials with Small Voids

Asymptotic formulae for the mechanical and electric fields in a piezoelectric body with a small void are derived and justified. Such results are new and useful for applications in the field of design of smart materials. In this way the topological derivatives of shape functionals are obtained for piezoelectricity. The asymptotic formulae are given in terms of the so-called polarization tensors (matrices), which are determined by the integral characteristics of voids. The distinguishing feature of the piezoelectricity boundary value problems under consideration is the absence of positive definiteness of a differential operator which is non-self-adjoint. Two specific Gibbs functionals of the problem are defined by the energy and the electric enthalpy. The topological derivatives are defined in different manners for each of the governing functionals. Actually, the topological derivative of the enthalpy functional is local, i.e., defined by the pointwise values of the mechanical and electric fields, which is contrary to the energy functional and some other suitable shape functionals which admit nonlocal topological derivatives, i.e., depending on the whole problem data. An example with weak interaction between mechanical and electric fields provides the analytic asymptotic expansions and can be used in numerical procedures of optimal design for smart materials.

Planar waveguide with “twisted” boundary conditions: Discrete spectrum

We consider a planar waveguide with combined Dirichlet and Neumann conditions imposed in a “twisted” way. We study the discrete spectrum and describe it dependence on the configuration of the boundary conditions. In particular, we show that in certain cases the model can have discrete eigenvalues emerging from the threshold of the essential spectrum. We give a criterium for their existence and construct them as convergent holomorphic series.

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.

Spectra of open waveguides in periodic media

Abstract We study the essential spectra of formally self-adjoint elliptic systems on doubly periodic planar domains perturbed by a semi-infinite periodic row of foreign inclusions. We show that the essential spectrum of the problem consists of the essential spectrum of the purely periodic problem and another component, which is the union of the discrete spectra of model problems in the infinite perturbation strip; these model problems arise by an application of the partial Floquet–Bloch–Gelfand transform.

Planar waveguide with “twisted” boundary conditions: Small width

We consider a planar waveguide with “twisted” boundary conditions. By twisting we mean a special combination of Dirichlet and Neumann boundary conditions. Assuming that the width of the waveguide goes to zero, we identify the effective (limiting) operator as the width of the waveguide tends to zero, establishes the uniform resolvent convergence in various possible operator norms, and gives the estimates for the rates of convergence. We show that studying the resolvent convergence can be treated as a certain threshold effect and we present an elegant technique which justifies such point of view.

Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary

The Neumann problem for the Poisson equation is considered in a domain Ωε ⊂ R with boundary components posed at a small distance ε > 0 so that in the limit, as ε → 0, the components touch each other at the point O with the tangency exponent 2m ≥ 2. Asymptotics of the solution uε and the Dirichlet integral ‖∇xuε; L(Ωε)‖ are evaluated and it is shown that main asymptotic term of uε and the existence of the finite limit of the integral depend on the relation between the spatial dimension n and the exponent 2m. For example, in the case n < 2m− 1 the main asymptotic term becomes of the boundary layer type and the Dirichlet integral has no finite limit. Some generalization are discussed and certain unsolved problems are formulated, in particular, non-integer exponents 2m and tangency of the boundary components along smooth curves. AMS subject classification: primary 35J25, secondary 46E35,35J20.

Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve

We consider an infinite planar straight strip perforated by small holes along a curve. In such domain, we consider a general second order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm resolvent convergence, we prove the convergence of the spectrum.