On entire solutions of the equations for the displacement fields in the deformation theory of plasticity with logarithmic hardening

Let $u\colon\mathbb R^2\to\mathbb R^2$ denote an entire solution of the homogeneous Euler–Lagrange equation associated to the energy used in the deformation theory of plasticity with logarithmic hardening. If $|u(x)|$ is of slower growth than $|x|$ as $|x|\to\infty$, then $u$ must be constant. Moreover we show that $u$ is affine if either $\sup_{\mathbb R^2}|\nabla u|<\infty$ or $\limsup_{|x|\to\infty}|x|^{-1}|u(x)|<\infty$.