  We are motivated by the study of the Microcanonical Variational Principle
within the Onsager's description of two-dimensional turbulence in the range of
energies where the equivalence of statistical ensembles fails. We obtain
sufficient conditions for the existence and multiplicity of solutions for the
corresponding Mean Field Equation on convex and "thin" enough domains in the
supercritical (with respect to the Moser-Trudinger inequality) regime. This is
a brand new achievement since existence results in the supercritical region
were previously known \un{only} on multiply connected domains. Then we study
the structure of these solutions by the analysis of their linearized problems
and also obtain a new uniqueness result for solutions of the Mean Field
Equation on thin domains whose energy is uniformly bounded from above. Finally
we evaluate the asymptotic expansion of those solutions with respect to the
thinning parameter and use it together with all the results obtained so far to
solve the Microcanonical Variational Principle in a small range of
supercritical energies where the entropy is eventually shown to be concave.
