Somigliana formula and the completeness of Papkovich-Neuber and Boussinesq-Galerkin solutions in elasticity (Q1106033)

In the absence of body forces, the displacement equations of equilibrium in the linear theory of homogeneous and isotropic elasticity are of the form \[ (1)\quad u_{i,ij}+\frac{1}{1-2\nu}u_{j,ji}=0, \] where \((u_ 1,u_ 2,u_ 3)\) is the displacement vector, \(\nu\) Poisson's ratio. The general solutions of Eqs. (1) have been established in many forms, among which the widely used ones may be the following Papkovich-Neuber solution \[ u_ i=4(1-\nu)H_ i-(H_ 0+x_ jH_ j),_ i;\quad \nabla^ 2H_ 0=0,\quad \nabla^ 2H_ i=0\quad (i=1,2,3) \] and Boussinesq- Galerkin solution \[ u_ i=2(1-\nu)\nabla^ 2B_ i-B_{1,1i};\quad \nabla^ 2\nabla^ 2B_ i=0\quad (i=1,2,3), \] where \(\nabla\) 2 is Laplacian operator. The completeness of above solutions has been studied e.g. by \textit{R. Mindlin} [Bull. Amer. Math. Soc. 42, 373-376 (1936)], \textit{E. Sternberg} and \textit{M. Gurtin} [Proc. 4th US Nath. Congr. Appl. Mech. 2, 793-797 (1962)], and \textit{M. Z. Wang} [Appl. Math. Modelling 12, No.3, 333-335 (1988) and J. Elasticity 15, 103-108 (1985; Zbl 0559.73025]. The purpose of the present paper is to give a new proof for the completeness of these solutions by connecting them with the Somigliana formula.