The authors consider the classical system of elastic equations \[ (\lambda +G)\frac{\partial \theta}{\partial x}+ G\nabla^2u +F_x=0,\quad (\lambda +G)\frac{\partial \theta}{\partial y}+ G\nabla^2v +F_y=0 \eqno(1) \] for an isotropic elastic body under boundary conditions \[ \alpha_{11}\sigma_{n1}+\beta_{11}u+\alpha_{21}\sigma_{n2}+\beta_{21}v= \psi^1,\quad \alpha_{12}\sigma_{n1}+\beta_{12}u+\alpha_{22}\sigma_{n2}+\beta_{22}v= \psi^2. \eqno(2) \] Here \(\alpha_{ij}\), \(\beta_{ij}\), \(\psi^i\), \(i,j=1,2\), are known functions at the boundary, \(\sigma_{nk}\), \(k=1,2\), are components of the stress tensor, \(u\) and \(v\) are the displacements. The displacements are represented by means of the Papkovitch–Neuber formulas as follows: \[ u =\Phi_1-\frac{1}{4(1-\mu)}\frac{\partial (\Phi_0 +x\Phi_1 +y \Phi_2)}{\partial x}, \quad v =\Phi_2-\frac{1}{4(1-\mu)}\frac{\partial (\Phi_0 +x\Phi_1 +y \Phi_2)}{\partial y}, \] where \(\Phi_s\), \(s=0,1,2\), are harmonic functions. The authors consider a set of polynomial solutions and try to find an approximate solution to the problem (1), (2) by means of this polynomial representation. As an example, they give a very special and simple solution to the Laplace equation.
{Any restrictions on the coefficients \(\alpha_{ij}\) and \(\beta_{ij}\) in the boundary conditions are absent, which leads, in general, to ill-posedness of the problem (1), (2). The mixed problem of the elasticity theory is, of course, a particular case of the problem (1), (2), but it cannot be solved by means of the polynomial representation in the whole region, including the boundary, since at points of change of boundary conditions the solution has, in general, oscillating singularities.}.