Sufficient Conditions for Global Minimality of Metastable States in a Class of Non-convex Functionals: A Simple Approach Via Quadratic Lower Bounds

Abstract

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 $v(x)$ for the 1D problem is automatically a global minimizer for the 2D problem on the asymmetric torus.

Appendix: Solving the Constrained Eigenvalue Problem

We seek to compute the minimizer of $\langle \psi , (B - 2v^{2})\psi \rangle$ subject to

$$\begin{aligned} \langle \psi ^{2} \rangle&= 1, \\ \langle \psi , \partial _{x} v(\mathbf {x}) \rangle&= 0. \end{aligned}$$

Due to the large matrices involved, it is more efficient to first ignore the orthogonality constraint and build a low rank approximation to $(B-2v^{2})$ over the entire unconstrained space. Once the low rank approximation is built, we then project out the appropriate orthogonality constraints and solve for the minimizer of $(B- 2v^{2})$.

• 1. Build numerical finite difference operators:

  $$\begin{aligned} \mathbf {L}&= \gamma ^{-2}(\varDelta ^{2}) - (\varDelta )(3v^{2}(\mathbf {x}) - 1) + I, \\ \mathbf {D}&= (-\varDelta ). \end{aligned}$$

• 2. Use MATLAB’s eigenvalue function

  $$\begin{aligned}{}[\mathbf {Q} \;\; \varvec{\Lambda } ] = eigs( \mathbf {L}, \; \mathbf {D}, \; k, \;\hbox {`}sm\hbox {'}) \end{aligned}$$

  to compute the first k eigenvectors and values to the eigenvalue problem:

  $$\begin{aligned} \mathbf {L} \mathbf {q}_{j}&= \lambda _{j} \mathbf {D} \mathbf {q}_{j} \quad j = 1,\ldots , k. \end{aligned}$$

  Here $\mathbf {Q} = [\mathbf {q}_{1} \; \mathbf {q}_{2}, \ldots , \mathbf {q}_{k}]$, while ${\varvec{\Lambda }} = {\hbox {diag}}[\lambda _{1} \; \lambda _{2},\ldots ,\lambda _k]$.

• 3. Orthonormalize $\mathbf {q}$ with respect to the standard $\ell ^{2}$ inner product (if they are not orthonormal already).

We now have the numerical representation for operator $(B-2v^{2})$ as follows:

$$\begin{aligned} (B - 2v^{2})&\approx \mathbf {D}^{-1} \mathbf {L} \\&= \mathbf {Q}{\varvec{\Lambda }}\mathbf {Q}^{T} + \mathbf {R}, \end{aligned}$$

where $\mathbf {R}^{T} = \mathbf {R}$ is a remainder term. Choose $k$ large enough so that $\mathbf {R}$ is a positive-definite matrix. Hence for any discrete vector $\psi$, we then have:

$$\begin{aligned} \psi ^{T} \mathbf {D}^{-1} \mathbf {L} \psi&\ge \psi ^{T} \mathbf {Q} \varvec{\Lambda }\mathbf {Q}^{T} \psi \quad {\hbox {for all }}\, \psi . \end{aligned}$$

Finally, we minimize $\psi ^{T} \mathbf {Q} {\varvec{\Lambda }} \mathbf {Q}^{T} \psi$ over the subspace $\psi ^{T} \mathbf {e}_{1} = 0$ with $||\psi || = 1$, where $\mathbf {e}_{1} \approx \frac{\partial v}{\partial x}$ is the discrete vector for $\frac{\partial v}{\partial x}$. The above problem can be solved using straightforward methods as $k$ will generally be small.