On certain boundary-value problems with strong vanishing on the boundary
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 with boundary , we have three equations to be satisfied by six stress components. The equations of equilibrium, coupled with conditions on , cannot determine the stress, but they do define a class of stress distributions, provided the body-force and the conditions on 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 in the sense that this body-force, and all its derivatives up to order , vanish on , then there exists a stress distribution which also vanishes strongly on , the order of vanishing being greater by one than that of the body-force.