Mathematical properties of free Gibbs energy I'm wandering, if some mathematical properties of the free Gibbs energy (also called thermodynamic potential) are well known.
For instance in chemistry, if $n_i^\alpha$ denotes the number of moles of species $i$ in phase $\alpha$ and G is the free Gibbs energy.
I know that G must be an homogeneous function of degree one in $n_i^\alpha$.

Do we have more information like continuity or coercivity properties of G ?
n is discontinuous. The least amount you can change it by is ~1 /6.022E23 of 1 mole. You'll have to be more specific about what type of (mathematical) continuity you are referring to, but it doesn't seem useful to ask about its fine grained continuity since it is a macroscopic concept.  Statistical (quantum) Thermodynamics is used both for microscopic systems and as the basis for macroscopic quantities such as Gibbs Free Energy. You need to be careful: GFE is one meaning of Thermodynamic Potential, but not the only one. I don't have any idea what you mean by G's coercivity.