Uniform convergence and asymptotics for problems in domains finely perforated along a prescribed manifold in the case of the homogenized Dirichlet condition

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F21%3A50019586" target="_blank" >RIV/62690094:18470/21:50019586 - isvavai.cz</a>

Výsledek na webu

<a href="https://iopscience.iop.org/article/10.1070/SM9435" target="_blank" >https://iopscience.iop.org/article/10.1070/SM9435</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1070/SM9435" target="_blank" >10.1070/SM9435</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Uniform convergence and asymptotics for problems in domains finely perforated along a prescribed manifold in the case of the homogenized Dirichlet condition

Popis výsledku v původním jazyce

A boundary value problem for a second-order elliptic equation with variable coefficients is considered in a multidimensional domain which is perforated by small holes along a prescribed manifold. Minimal natural conditions are imposed on the holes. In particular, all of these are assumed to be of approximately the same size and have a prescribed minimal distance to neighbouring holes, which is also a small parameter. The shape of the holes and their distribution along the manifold are arbitrary. The holes are divided between two sets in an arbitrary way. The Dirichlet condition is imposed on the boundaries of holes in the first set and a nonlinear Robin boundary condition is imposed on the boundaries of holes in the second. The sizes and distribution of holes with the Dirichlet condition satisfy a simple and easily verifiable condition which ensures that these holes disappear after homogenization and a Dirichlet condition on the manifold in question arises instead. We prove that the solution of the perturbed problem converges to the solution of the homogenized one in the W 1/2-norm uniformly with respect to the right-hand side of the equation, and an estimate for the rate of convergence that is sharp in order is deduced. The full asymptotic solution of the perturbed problem is also constructed in the case when the holes form a periodic set arranged along a prescribed hyperplane. Bibliography: 32 titles.

Název v anglickém jazyce

Uniform convergence and asymptotics for problems in domains finely perforated along a prescribed manifold in the case of the homogenized Dirichlet condition

Popis výsledku anglicky

A boundary value problem for a second-order elliptic equation with variable coefficients is considered in a multidimensional domain which is perforated by small holes along a prescribed manifold. Minimal natural conditions are imposed on the holes. In particular, all of these are assumed to be of approximately the same size and have a prescribed minimal distance to neighbouring holes, which is also a small parameter. The shape of the holes and their distribution along the manifold are arbitrary. The holes are divided between two sets in an arbitrary way. The Dirichlet condition is imposed on the boundaries of holes in the first set and a nonlinear Robin boundary condition is imposed on the boundaries of holes in the second. The sizes and distribution of holes with the Dirichlet condition satisfy a simple and easily verifiable condition which ensures that these holes disappear after homogenization and a Dirichlet condition on the manifold in question arises instead. We prove that the solution of the perturbed problem converges to the solution of the homogenized one in the W 1/2-norm uniformly with respect to the right-hand side of the equation, and an estimate for the rate of convergence that is sharp in order is deduced. The full asymptotic solution of the perturbed problem is also constructed in the case when the holes form a periodic set arranged along a prescribed hyperplane. Bibliography: 32 titles.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

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Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

SBORNIK MATHEMATICS

ISSN

1064-5616

e-ISSN

1468-4802

Svazek periodika

212

Číslo periodika v rámci svazku

8

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

54

Strana od-do

1068-1121

Kód UT WoS článku

000707456500001

EID výsledku v databázi Scopus

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