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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains


Author: Jeff E. Lewis
Journal: Trans. Amer. Math. Soc. 320 (1990), 53-76
MSC: Primary 35J25; Secondary 35Q20, 35S05, 45K05, 47A53, 47G05, 73C02, 76D07
MathSciNet review: 1005935
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Abstract: The symbolic calculus of pseudodifferential operators of Mellin type is applied to study layer potentials on a plane domain $ {\Omega ^ + }$ whose boundary $ {\partial\Omega ^ + }$ is a curvilinear polygon. A "singularity type" is a zero of the determinant of the matrix of symbols of the Mellin operators and can be used to calculate the "bad values" of $ p$ for which the system is not Fredholm on $ {L^p}(\partial {\Omega ^ + })$.

Using the method of layer potentials we study the singularity types of the system of elastostatics

$\displaystyle L{\mathbf{u}} = \mu \Delta {\mathbf{u}} + (\lambda + \mu )\nabla \operatorname{div} {\mathbf{u}} = 0.$

in a plane domain $ {\Omega ^ + }$ whose boundary $ {\partial\Omega ^ + }$ is a curvilinear polygon. Here $ \mu > 0$ and $ -\mu \le \lambda \le +\infty$. When $ \lambda = +\infty$, the system is the Stokes system of hydrostatics. For the traction double layer potential, we show that all singularity types in the strip $ 0 < \operatorname{Re} z < 1$ lie in the interval $ \left( {\frac{1} {2},1} \right)$ so that the system of integral equations is a Fredholm operator of index 0 on $ {L^p}(\partial {\Omega ^ + })$ for all $ p$, $ 2 \le p < \infty $. The explicit dependence of the singularity types on $ \lambda$ and the interior angles $ \theta$ of $ {\partial\Omega ^ + }$ is calculated; the singularity type of each corner is independent of $ \lambda$ iff the corner is nonconvex.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-1005935-5
PII: S 0002-9947(1990)1005935-5
Article copyright: © Copyright 1990 American Mathematical Society