Sufficient Conditions for Global Minimality of Metastable States in a Class of Non-Convex Functionals: a Simple Approach via Convex Lower Bounds

We consider mass-constrained minimizers for a class of non-convex energy functionals involving double-well potentials. Based upon global convex lower bounds to the energy, we introduce a simple strategy to find sufficient conditions on a given critical point (metastable state) to be a global minimizer. We show that this strategy works well for the one exact and known metastable state: the constant state. In doing so, we numerically derive an almost optimal lower bound for both the order-disorder transition curve of the Ohta-Kawasaki energy and the liquid-solid interface of the Phase Field Crystal energy. We discuss how this strategy extends to non-constant computed metastable states, and the resulting symmetry issues which one must overcome. We give a preliminary analysis of these symmetry issues by addressing the global optimality of a computed lamellar structure for the Ohta-Kawasaki energy in one (1D) and two (2D) space dimensions, and on a two dimensional, asymmetric, flat torus. We also consider global optimality of a non-constant state for a spatially inhomogenous perturbation of the Ohta-Kawasaki energy. Finally we use one of our simple convex lower bounds to rigorously prove that for certain values of the Ohta-Kawasaki parameter and aspect ratio of an asymmetric flat torus, any global minimizer v(x) for the 1D problem is automatically a global minimizer for the 2D problem on the asymmetric torus.