Lets consider a steady boundary value problem of linear elasticity, subjected to a mix of Dirichlet and Neumann boundary conditions.
The domain boundary consists of 4 linear segments and 4 circular segments (see "Domain" in the linked picture). I apply Dirichlet conditions (0,0) to all linear segments and constant surface loading (1,1) on circular segments. Now I solve the problem and obtain expected symmetric displacement field, horizontal component (see "original solution"). Now, I change the surface loading on bottom right circular segment to (0,0), and the displacement field changes accordingly (see "zero surface stress").
Now, for the final test, I remove all conditions, i.e., let the displacement field vary and surface stress vary from the bottom right circular segment, and I obtain different displacement field (see "undefined solution").
The linear elasticity problems have been solved using FreeFem++. I have tested, that both iterative solvers (GMRES, tolerance 1.0e-10) and direct solvers (LU, UMFPACK sparse solver) give the same solution, therefore it seems that the solution is unique.
Question: what sets this solution to be unique?
It certainly is not remaining boundary conditions, since we have already obtained 3 different solutions using them, and there are probably infinite amount more. Could it be some sort of lowest elastic energy solution of the given problem, somehow arising from Galerkin formulation? Is there some literature I could read on how that could work?
Here is the illustration picture:
http://www.mech.kth.se/~ugis/img/test_res_unique_allsol.png
Here are FreeFem++ scripts, used to generate shown solutions, and also all individual solutions: http://www.mech.kth.se/~ugis/edp/