Integral Methods in Science and Engineering by Unknown

Author:Unknown
Language: eng
Format: epub, pdf
ISBN: 9783030160777
Publisher: Springer International Publishing

Paolo Musolino

Email: [email protected]

Julia Orlik

Email: [email protected]

19.1 Introduction

We develop tools, which can be useful for elliptic boundary value problems on domains with a periodic structure with holes involving some linear or non-linear Robin-type conditions on the oscillating interface [GrEtAl12, KhEtAl17, GaEtAl16], or contact problems (see [GaMe18, GrOr18]).

This paper considers rescaling of functions from the Bessel potential, Riesz potential, and Sobolev–Slobodetskii spaces on the boundary or in the domain and also rescaling of the boundary trace operator.

Denote by Ω a bounded domain in with Lipschitz boundary. Let Y := (0, 1)n be the reference cell. We denote by T a hole, that is, an open set, which closure is strictly included in Y and let (see Fig. 19.1). Let ∂T be the Lipschitz boundary of T and ν be the outward to Y ∗ unit normal vector on the boundary ∂T. Recall, e.g., from [CiEtAl12] that in the periodic setting, every point can be written as Here the integer function for a vector means the floor function ⌊⋅⌋ for each of its components. Denote i.e., the set Λ ε contains the parts of the cells intersecting the boundary ∂Ω.

Fig. 19.1Bounded domain with periodically distributed holes

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