Layer Potential Techniques
Layer Potential Techniques
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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