1,721,023 research outputs found
Singular perturbation and homogenization problems in a periodically perforated domain. A functional analytic approach (Tesi di dottorato, Università degli Studi di Padova, Italia)
No full textProblemi di perturbazione singolare e di omogeneizzazione in un dominio periodicamente perforato. Un approccio funzionale analitco
No full textA Functional Analytic Approach for a Singularly Perturbed Dirichlet Problem for the Laplace Operator in a Periodically Perforated Domain
Get PDFWe consider a sufficiently regular bounded open connected subset of such that and such that \mathbb{R}^n \setminus \cl\Omega is connected. Then we choose a point . If is a small positive real number, then we define the periodically perforated domain T(\epsilon) \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\cl(w+\epsilon \Omega +z). For each small positive , we introduce a particular Dirichlet problem for the Laplace operator in the set . More precisely, we consider a Dirichlet condition on the boundary of the set , and we denote the unique periodic solution of this problem by . Then we show that (suitable restrictions of) can be continued real analytically in the parameter around
Domain Perturbation for the Solution of a Periodic Dirichlet Problem
Get PDFWe prove that the solution of the periodic Dirichlet problem for the Laplace equation depends real analytically on a suitable parametrization of the shape of the domain, on the periodicity parameters, and on the Dirichlet datum
A mixed problem for the Laplace operator in a domain with moderately close holes
Get PDFWe investigate the behavior of the solution of a mixed problem in a domain with two moderately close holes. We introduce a positive parameter ε and we define a perforated domain Ωε obtained by making two small perforations in an open set. Both the size and the distance of the cavities tend to 0 as ε → 0. For ε small, we denote by uε the solution of a mixed problem for the Laplace equation in Ωε. We describe what happens to uε as ε → 0 in terms of real analytic maps and we compute an asymptotic expansion
A singularly perturbed nonlinear traction problem in a periodically perforated domain: A functional analytic approach
No full textWe consider a periodically perforated domain obtained by making in a periodic set of holes, each of them of size proportional to . Then we introduce a nonlinear boundary value problem for the Lam\'e equations in such a periodically perforated domain. The unknown of the problem is a vector valued function which represents the displacement attained in the equilibrium configuration by the points of a periodic linearly elastic matrix with a hole of size contained in each periodic cell. We assume that the traction exerted by the matrix on the boundary of each hole depends (nonlinearly) on the displacement attained by the points of the boundary of the hole. Then our aim is to describe what happens to the displacement vector function when tends to . Under suitable assumptions we prove the existence of a family of solutions with a prescribed limiting behaviour when approaches . Moreover, the family is in a sense locally unique and can be continued real analytically for negative values of
Real analytic families of harmonic functions in a planar domain with a small hole
Get PDFWe consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter and we denote by the corresponding solution. The behavior of for small and positive can be described in terms of real analytic functions of two variables evaluated at . We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by for small and describe by real analytic functions of . Then it is natural to ask what happens when is negative. The case of boundary data depending on is also considered. The aim is to study real analytic families of harmonic functions which are not necessarily solutions of a particular boundary value problem
Moderately close Neumann inclusions for the Poisson equation
Get PDFWe investigate the behavior of the solution of a mixed problem for the Poisson equation in a domain with two moderately close holes. If ρ1 and ρ2 are two positive parameters, we define a perforated domain Ω(ρ1,ρ2) by making two small perforations in an open set: the size of the perforations is ρ1ρ2, while the distance of the cavities is proportional to ρ1. Then, if r∗ is small enough, we analyze the behavior of the solution for (ρ1,ρ2) close to the degenerate pair (0,r∗)
Asymptotic behavior of the longitudinal permeability of a periodic array of thin cylinders
Get PDFWe consider a Newtonian fluid flowing at low Reynolds numbers
along a spatially periodic array of cylinders of diameter proportional
to a small nonzero parameter . Then for and
close to we denote by the longitudinal permeability.
We are interested in studying the asymptotic behavior of
as tends to . We analyze for
close to by an approach based on functional analysis and potential theory,
which is alternative to that of asymptotic analysis. We prove that
can be written as the sum of a logarithmic term and a
power series in . Then, for small , we provide an
asymptotic expansion of the longitudinal permeability in terms of the sum
of a logarithmic function of the square of the capacity of the cross section
of the cylinders and a term which does not depend of the shape of the unit
inclusion (plus a small remainder)
A nonlinear problem for the Laplace equation with a degenerating Robin condition
Get PDFWe investigate the behavior of the solutions of a mixed problem for the Laplace equation in a domain Ω. On a part of the boundary ∂Ω, we consider a Neumann condition, whereas in another part, we consider a nonlinear Robin condition, which depends on a positive parameter δ in such a way that for δ = 0 it degenerates into a Neumann condition. For δ small and positive, we prove that the boundary value problem has a solution u(δ,·). We describe what happens to u(δ,·) as δ→0 by means of representation formulas in terms of real analytic maps. Then, we confine ourselves to the linear case, and we compute explicitly the power series expansion of the solution
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