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Embeddings for the space $LD_\gamma ^{p}$ on sets of finite perimeter

Published online by Cambridge University Press:  07 May 2019

Nikolai V. Chemetov  and
Anna L. Mazzucato
Nikolai V. Chemetov
Affiliation:
Departamento de Matemática, Universidade Federal do Amazonas, Av. Rodrigo Octávio 6200, 69080-900Manaus, Amazonas, Brazil ([email protected])
Anna L. Mazzucato
Affiliation:
Penn State University, University Park, PA16802, USA ([email protected])
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Abstract

Given an open set with finite perimeter $\Omega \subset {\open R}^n$, we consider the space $LD_\gamma ^{p}(\Omega )$, $1\les p<\infty $, of functions with pth-integrable deformation tensor on Ω and with pth-integrable trace value on the essential boundary of Ω. We establish the continuous embedding $LD_\gamma ^{p}(\Omega )\subset L^{pN/(N-1)}(\Omega )$. The space $LD_\gamma ^{p}(\Omega )$ and this embedding arise naturally in studying the motion of rigid bodies in a viscous, incompressible fluid.

Keywords

Finite perimeterintegrable deformationtraceessential boundarySobolev embeddingfluid–structure interaction
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2442 - 2461
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Copyright
Copyright © Royal Society of Edinburgh 2019

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