Search results for "singularly perturbed perforated domain"

showing 4 items of 4 documents

A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

2016

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain $\Omega_{\boldsymbol\varepsilon}$ obtained by making a small hole of size $\varepsilon_1 \varepsilon_2 $ in an open regular subset $\Omega$ of $\mathbb{R}^n$ at distance $\varepsilon_1$ from the boundary $\partial\Omega$. As $\varepsilon_1 \to 0$, the perforation shrinks to a point and, at the same time, approaches the boundary. When $\boldsymbol\varepsilon \to (0,0)$, the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by $u_{\bolds…

Asymptotic analysisGeneral MathematicsBoundary (topology)Asymptotic expansion01 natural sciences35J25; 31B10; 45A05; 35B25; 35C20Mathematics - Analysis of PDEsSettore MAT/05 - Analisi MatematicaFOS: Mathematics[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]Mathematics (all)Mathematics - Numerical Analysis0101 mathematicsMathematicsDirichlet problemLaplace's equationDirichlet problemAnalytic continuationApplied Mathematics010102 general mathematicsMathematical analysisHigh Energy Physics::PhenomenologyReal analytic continuation in Banach spaceNumerical Analysis (math.NA)Physics::Classical Physics010101 applied mathematicsasymptotic analysisLaplace operatorPhysics::Space PhysicsAsymptotic expansion; Dirichlet problem; Laplace operator; Real analytic continuation in Banach space; Singularly perturbed perforated domain; Mathematics (all); Applied MathematicsAsymptotic expansionLaplace operator[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]Singularly perturbed perforated domainAnalytic functionAnalysis of PDEs (math.AP)Asymptotic expansion; Dirichlet problem; Laplace operator; Real analytic continuation in Banach space; Singularly perturbed perforated domain;
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Moderately close Neumann inclusions for the Poisson equation

2016

We 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∗). Copyright © 2016 John Wiley & Sons, Ltd.

General Mathematics010102 general mathematicsMathematical analysisGeneral Engineeringmixed problem; moderately close holes; Poisson equation; real analytic continuation in Banach space; singularly perturbed perforated domain; Mathematics (all); Engineering (all)Poisson equation01 natural sciences010101 applied mathematicsmixed problemsingularly perturbed perforated domainEngineering (all)Settore MAT/05 - Analisi MatematicaMathematics (all)0101 mathematicsPoisson's equationmoderately close holesMathematicsreal analytic continuation in Banach space
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Local uniqueness of the solutions for a singularly perturbed nonlinear nonautonomous transmission problem

2020

Abstract We consider the Laplace equation in a domain of R n , n ≥ 3 , with a small inclusion of size ϵ . On the boundary of the inclusion we define a nonlinear nonautonomous transmission condition. For ϵ small enough one can prove that the problem has solutions. In this paper, we study the local uniqueness of such solutions.

Local uniqueness of the solutionsLaplace's equation020502 materialsApplied MathematicsNonlinear nonautonomous transmission problem010102 general mathematicsMathematical analysisA domainBoundary (topology)02 engineering and technology01 natural sciencesNonlinear systemMathematics - Analysis of PDEs35J25 31B10 35J65 35B25 35A020205 materials engineeringTransmission (telecommunications)Settore MAT/05 - Analisi MatematicaLocal uniqueness of the solutions; Nonlinear nonautonomous transmission problem; Singularly perturbed perforated domainFOS: MathematicsUniqueness0101 mathematicsSingularly perturbed perforated domainAnalysisMathematicsAnalysis of PDEs (math.AP)
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Global representation and multiscale expansion for the Dirichlet problem in a domain with a small hole close to the boundary

2019

For each pair (Formula presented.) of positive parameters, we define a perforated domain (Formula presented.) by making a small hole of size (Formula presented.) in an open regular subset (Formula presented.) of (Formula presented.) ((Formula presented.)). The hole is situated at distance (Formula presented.) from the outer boundary (Formula presented.) of the domain. Thus, when (Formula presented.) both the size of the hole and its distance from (Formula presented.) tend to zero, but the size shrinks faster than the distance. Next, we consider a Dirichlet problem for the Laplace equation in the perforated domain (Formula presented.) and we denote its solution by (Formula presented.) Our ai…

multiscale asymptotic expansionmulti-scale asymptotic expansionBoundary (topology)01 natural sciences35J25; 31B10; 45A05; 35B25; 35C20Domain (mathematical analysis)Settore MAT/05 - Analisi MatematicaSituated[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]Dirichlet problem; Laplace operator; multiscale asymptotic expansion; real analytic continuation in Banach space; singularly perturbed perforated domainSmall hole[MATH]Mathematics [math]0101 mathematicsRepresentation (mathematics)MathematicsDirichlet problemDirichlet problemApplied Mathematics010102 general mathematicsMathematical analysisA domain010101 applied mathematicssingularly perturbed perforated domainLaplace operatorLaplace operatorAnalysisreal analytic continuation in Banach space
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