The paper is inspired by the GNN theorem stating that in an
n-dimensional ball, all the solutions u of the f-inhomogeneous
Laplace equation (with Dirichlet boundary conditions) are radially
symmetric for all the reasonable f. The text pays attention to the
similar (approximate) inheritance of the symmetry for the
discretized versions of the same equation (using cubic mesh of
size h). The authors start from the reminder of an elementary n=1
counterexample (one can only expect an approximate inheritance of
the symmetry) and they offer some precise results contrasting the
defects O(h) and O(1/|log h|) of symmetry at n=1 and n>1.


MR2332550 McKenna, P. J.; Reichel, W.: Gidas-Ni-Nirenberg results
for finite difference equations: estimates of approximate
symmetry. J. Math. Anal. Appl. 334 (2007), no. 1, 206--222. 39A12
(35J60)