To be concrete, let us consider the energy $U$ as the relevant fundamental relation. To be even more specific, let's take a simple, single-component system: $U = U(S,V,N)$. One can then show that stability for the given system corresponds to this fundamental relation being everywhere above its tangent planes; that is, to its being a convex function. This notion of stability is referred to as global stability. There is another notion of stability, known as local stability, which refers to the Hessian of $U$ being such that each of its principal submatrices are positive definite. Essentially, this is a requirement about the "local convexity" of $U$.
How are global and local stability related, and under what hypotheses are they equivalent? As far as I can tell, one can have local stability without global stability, but the converse need not hold. Can we say something sensible about local stability implying global stability if local stability holds everywhere?
As an aside, this question is asked in reference to the following problem posed by Callen:
If $g$, the molar Gibbs function, is a convex function of $x_1,...x_{r-1}$, show that a change of variables to $x_2,...x_r$ results in $g$ being a convex function of $x_2,...x_r$. That is, show that the convexity condition of the molar Gibbs potential is independent of the choice of the "redundant" mole fraction.
I can imagine how to answer this in terms of local stability just by using the chain rule (for derivatives), but it seems a lot less tractable to answer in terms of global stability. I was therefore wondering about the above so as to answer the problem indirectly.
