The construction of periodic unfolding operators on some compact Riemannian manifolds

Sören Dobberschütz; Michael Böhm

doi:10.1515/apam-2013-0013

The construction of periodic unfolding operators on some compact Riemannian manifolds

Sören Dobberschütz, Michael Böhm

Department of Chemistry

1 Citation (Scopus)

Abstract

The notion of periodic unfolding has become a standard tool in the theory of periodic homogenization. However, all the results obtained so far are only applicable to the "flat" Euclidean space R ⁿ. In this paper, we present a generalization of the method of periodic unfolding applicable to structures defined on certain compact Riemannian manifolds. While many results known from unfolding in domains of R ⁿ can be recovered, for the unfolding of gradients a transport operator has to be defined. This operator connects vector fields on the manifold and in the reference cell, which allows for the formulation of general two-scale problems. We illustrate the use of the new unfolding technique with a simple elliptic model-problem.

Original language	English
Journal	Advances in Pure and Applied Mathematics
Volume	5
Issue number	1
Pages (from-to)	31-45
Number of pages	15
ISSN	1867-1152
DOIs	https://doi.org/10.1515/apam-2013-0013
Publication status	Published - 2014

Keywords

homogenization
Periodic unfolding
Riemannian manifolds

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AU - Dobberschütz, Sören

AU - Böhm, Michael

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N2 - The notion of periodic unfolding has become a standard tool in the theory of periodic homogenization. However, all the results obtained so far are only applicable to the "flat" Euclidean space R n. In this paper, we present a generalization of the method of periodic unfolding applicable to structures defined on certain compact Riemannian manifolds. While many results known from unfolding in domains of R n can be recovered, for the unfolding of gradients a transport operator has to be defined. This operator connects vector fields on the manifold and in the reference cell, which allows for the formulation of general two-scale problems. We illustrate the use of the new unfolding technique with a simple elliptic model-problem.

AB - The notion of periodic unfolding has become a standard tool in the theory of periodic homogenization. However, all the results obtained so far are only applicable to the "flat" Euclidean space R n. In this paper, we present a generalization of the method of periodic unfolding applicable to structures defined on certain compact Riemannian manifolds. While many results known from unfolding in domains of R n can be recovered, for the unfolding of gradients a transport operator has to be defined. This operator connects vector fields on the manifold and in the reference cell, which allows for the formulation of general two-scale problems. We illustrate the use of the new unfolding technique with a simple elliptic model-problem.

KW - homogenization

KW - Periodic unfolding

KW - Riemannian manifolds

U2 - 10.1515/apam-2013-0013

DO - 10.1515/apam-2013-0013

M3 - Journal article

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SP - 31

EP - 45

JO - Advances in Pure and Applied Mathematics

JF - Advances in Pure and Applied Mathematics

IS - 1

ER -

The construction of periodic unfolding operators on some compact Riemannian manifolds

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