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Mathematical Methods in Elasticity Imaging$
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Habib Ammari, Elie Bretin, Josselin Garnier, Hyeonbae Kang, Hyundae Lee, and Abdul Wahab

Print publication date: 2015

Print ISBN-13: 9780691165318

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691165318.001.0001

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Layer Potential Techniques

Layer Potential Techniques

(p.4) Chapter One Layer Potential Techniques
Mathematical Methods in Elasticity Imaging

Habib Ammari

Elie Bretin

Josselin Garnier

Hyeonbae Kang

Hyundae Lee

Abdul Wahab

Princeton University Press

This chapter considers some well-known results on the solvability and layer potentials for static and time-harmonic elasticity equations. It first reviews commonly used function spaces before introducing equations of linear elasticity and decomposing the displacement field into the sum of an irrotational (curl-free) and a solenoidal (divergence-free) field using the Helmholtz decomposition theorem. It then discusses the radiation condition for the time-harmonic elastic waves, which is used to select the physical solution to exterior problems. It also describes the layer potentials associated with the operators of static and time-harmonic elasticity, along with their mapping properties, and proves decomposition formulas for the displacement fields. Finally, it derives the Helmholtz–Kirchhoff identities, analyzes Neumann and Dirichlet functions, and states a generalization of Meyer's theorem concerning the regularity of solutions to the equations of linear elasticity.

Keywords:   layer potential, elasticity equation, function space, linear elasticity, displacement field, Helmholtz decomposition theorem, radiation condition, elastic wave, Helmholtz–Kirchhoff identities, Dirichlet function

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