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where A = d2/Sx2 + d2/dy2 , R is a plane domain with boundary C, and d/dn denotes differentiation in the direction of the outer normal to C. Assuming the existence and uniqueness of the solution of (1), we developed a method for obtaining upper and lower bounds for w(x0, y0), the point x0 , yn of R being given in advance. The upper and lower bounds were given in terms of integrals of two classes of functions, one satisfying certain "boundary conditions and the other certain partial differential equations. In the present paper we shall be concerned with the application of the same method to a boundary value problem involving a system of partial differential equations. For definiteness we shall deal only with the first boundary value problem of three dimensional elasticity, i.e.

Upper and lower bounds for the solution of the first boundary value problem of elasticity