Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On certain boundary-value problems with strong vanishing on the boundary


Authors: J. McCrea and J. L. Synge
Journal: Quart. Appl. Math. 24 (1967), 355-364
MSC: Primary 53.45; Secondary 35.00
DOI: https://doi.org/10.1090/qam/216414
MathSciNet review: 216414
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Abstract: In Newtonian statics of a continuum we have a symmetric stress tensor with six components and a body-force with three components, and the divergence of the stress tensor equals the body-force vector reversed. If the body-force vector as assigned in some finite domain $ I$ with boundary $ B$, we have three equations to be satisfied by six stress components. The equations of equilibrium, coupled with conditions on $ B$, cannot determine the stress, but they do define a class of stress distributions, provided the body-force and the conditions on $ B$ are consistent. The purpose of this paper is to show that, if the body-force satisfies the usual conditions of equilibrium and vanishes strongly on $ B$ in the sense that this body-force, and all its derivatives up to order $ N$, vanish on $ B$, then there exists a stress distribution which also vanishes strongly on $ B$, the order of vanishing being greater by one than that of the body-force.


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DOI: https://doi.org/10.1090/qam/216414
Article copyright: © Copyright 1967 American Mathematical Society


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