For geotechnical engineering problems, it is common to fix a single component of displacement along a boundary as a Dirichlet boundary condition (roller boundary condition). However, I'm having trouble seeing why this leads to a well-posed problem.
The number of unknowns in an elastic problem is equal to the dimension of the problem (i.e. one unknown for each displacement component for the Navier equations). I was under the impression you needed to specify a boundary condition for each unknown in a boundary value problem. Why can we get away with a single boundary condition for a single component of displacement? Is there an implicit stress boundary condition implied when we do this? I've included an example sketch with a common configuration. My intuition says this should have a unique solution but I can't see why mathematically.