TY - JOUR
T1 - Sufficient Conditions for Global Minimality of Metastable States in a Class of Non-convex Functionals
T2 - A Simple Approach Via Quadratic Lower Bounds
AU - Shirokoff, David
AU - Choksi, Rustum
AU - Nave, Jean Christophe
N1 - Funding Information:
We are indebted to one of the referees for their detailed comments that substantially improved the article. The authors would like to thank Ben Mares, Cyrill Muratov, Leo Stein, Marco Veneroni, and Thomas Wanner for useful conservations, comments, and suggestions. This research was partly supported by NSERC (Canada) Discovery Grants. J.-C.N. was also supported by an NSERC Accelerator Supplement.
Publisher Copyright:
© 2015, Springer Science+Business Media New York.
PY - 2015/6/1
Y1 - 2015/6/1
N2 - We consider mass-constrained minimizers for a class of non-convex energy functionals involving a double-well potential. Based upon global quadratic 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 that 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. We also consider global optimality of a non-constant state for a spatially in-homogenous perturbation of the 2D Ohta–Kawasaki energy. Finally we use one of our simple quadratic lower bounds to rigorously prove that for certain values of the Ohta–Kawasaki parameter and aspect ratio of an asymmetric torus, any global minimizer $$v(x)$$v(x) for the 1D problem is automatically a global minimizer for the 2D problem on the asymmetric torus.
AB - We consider mass-constrained minimizers for a class of non-convex energy functionals involving a double-well potential. Based upon global quadratic 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 that 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. We also consider global optimality of a non-constant state for a spatially in-homogenous perturbation of the 2D Ohta–Kawasaki energy. Finally we use one of our simple quadratic lower bounds to rigorously prove that for certain values of the Ohta–Kawasaki parameter and aspect ratio of an asymmetric torus, any global minimizer $$v(x)$$v(x) for the 1D problem is automatically a global minimizer for the 2D problem on the asymmetric torus.
KW - Convex/quadratic lower bound
KW - Double-well potential
KW - Global minimizers
KW - Metastable state
KW - Non-convex energy
KW - Ohta–Kawasaki functional
KW - Phase-field crystal functional
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U2 - 10.1007/s00332-015-9234-0
DO - 10.1007/s00332-015-9234-0
M3 - Article
AN - SCOPUS:84939941131
SN - 0938-8974
VL - 25
SP - 539
EP - 582
JO - Journal of Nonlinear Science
JF - Journal of Nonlinear Science
IS - 3
ER -