Existence and uniqueness of the critical wave number and critical Rayleigh number for the asymmetric planar Benard problem and a function space formulation of the nonlinear problem

We consider the Planar Benard Problem on the hexagonal lattice with "rigid-free" boundary conditions. Using analytic and numerical techniques, the existence and uniqueness of the critical wave number and the critical Rayleigh number for the problem are proved. Further, we consider function spaces required for Lyapunov-Schmidt reduction, and provide a framework which should lead to a proof of the existence and uniqueness of the standard family of solutions of the problem.