1,720,967 research outputs found
On the spectral stability of polyharmonic operators on singularly perturbed domains
Full text linkIn this thesis, we analyse the spectral convergence properties of higher order elliptic differential operators subject to singular domain perturbations and non-Dirichlet boundary conditions, with special attention to polyharmonic operators. We identify suitable conditions on the shape of the initial domain, on the shape of the perturbed domains, and on the geometry of the perturbation in order to assure the spectral stability. We find the limiting differential problem depending on the type of domain perturbation and the geometrical parameters governing the shape deformation of the initial domain. We prove that, under suitable conditions, the eigenvalues and the eigenprojections of the given differential operator in the perturbed domain converge to the eigenvalues and the eigenprojections of the limiting differential operator in the unperturbed domain. Finally, we prove convergence of the resolvent operators in the framework of the compact convergence of linear operators in Hilbert spaces.
More specifically, we first analyse the spectral convergence of a family of higher order self-adjoint elliptic operators subject to intermediate boundary conditions on perturbed domains defined locally by the hypographs of given functions. We prove a spectral stability theorem for this family of operators under the assumption that the convergence of the functions describing the boundary of the domain is sufficiently regular. Then we apply the theorem to study the spectral behaviour of polyharmonic operators with intermediate boundary conditions when the boundary of the domain undergoes a perturbation of oscillatory type, by adapting techniques introduced by J.M. Arrieta and P.D. Lamberti for the biharmonic operator. We prove that the limiting differential problem depends on the ratio between the amplitude and the period of the oscillation. Indeed there is a critical threshold above which there is spectral stability; that is, the eigenvalues and the eigenprojections of the perturbed problem converge to the corresponding eigenvalues and eigenprojections of the same differential problem in the limiting domain. Instead, under that threshold there is a different behaviour depending on the order of the polyharmonic operator and on the type of intermediate boundary conditions imposed at the boundary. If the ratio assumes exactly the critical value, then the limiting differential problem exhibits a strange boundary condition, which is characterized in terms of an auxiliary function satisfying a suitable differential problem. In order to treat this critical case we use homogenization techniques and macroscopic-microscopic decompositions, inspired by arguments used by J. Casado-Diaz and collaborators.
Then we consider the biharmonic operator and the Reissner-Mindlin 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 the spirit of the articles by J.M. Arrieta and collaborators for the Neumann Laplacian. In applications to linear elasticity, the operators under consideration are related to the deformation of a free elastic plate, a part of which shrinks to a segment. In contrast to the classical case of the Laplace operator, 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
Can dissipative Maxwell systems have open disks of eigenvalues?
No full textThe largest essential spectrum sigma e5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} of a transmission problem for the dissipative Maxwell system through a semi-transparent Faraday layer Sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} consists of the union of the Weyl essential spectra of a Maxwell operator with homogeneous boundary conditions and a carefully defined 'extended' essential spectrum of a pseudodifferential operator of order 0 on Sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}. However, only partial information is available regarding sigma e4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}. A natural question is then to clarify whether sigma e4=sigma e5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}. The open problem is to prove or disprove the existence of open disks of eigenvalues
On the spectral instability for weak intermediate triharmonic problems
Full text linkWe define the weak intermediate boundary conditions for the triharmonic operator −Δ3. We analyse the sensitivity of this type of boundary conditions upon domain perturbations. We construct a perturbation (Ωϵ)ϵ > 0 of a smooth domain Ω of
On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set
Full text linkLet be a bounded domain in with smooth boundary
, and let be the set of points in whose
distance from the boundary is smaller than . We prove that the eigenvalues
of the biharmonic operator on with Neumann boundary conditions
converge to the eigenvalues of a limiting problem in the form of system of
differential equations on
Spectral convergence analysis for the Reissner-Mindlin system in any dimension
Full text linkWe establish the convergence of the resolvent of the Reissner-Mindlin system in any dimension N≥2, with any of the physically relevant boundary conditions, to the resolvent of the biharmonic operator with suitably defined boundary conditions in the vanishing thickness limit. Moreover, given a thin domain Ωδ in RN with 1≤
Boundary homogenization for a triharmonic intermediate problem
Full text linkWe consider the triharmonic operator subject to homogeneous boundary conditions of intermediate type on a bounded domain of the N‐dimensional Euclidean space. We study its spectral behaviour when the boundary of the domain undergoes a perturbation of oscillatory type. We identify the appropriate limit problems that depend on whether the strength of the oscillation is above or below a critical threshold. We analyse in detail the critical case that provides a typical homogenization problem leading to a strange boundary term in the limit problem
Singular perturbation Dirichlet problem in a double-periodic perforated plane
No full textWe show that the spectrum of the Dirichlet problem for the Laplace operator -Δ in the plane R2 perforated by a double-periodic family of holes contains any a priori number of gaps, for sufficiently large holes. While this result was already known in the case of circular holes, we consider here a more general geometric setting with holes of the shape (Formula Presented.)
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Spectral Analysis of the Biharmonic Operator Subject to Neumann Boundary Conditions on Dumbbell Domains
Full text linkWe 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
- …
