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Numerics and analysis of Cahn--Hilliard critical points

Tobias Grafke, Sebastian Scholtes, Alfred Wagner, Maria G. Westdickenberg

TL;DR

The paper addresses the existence and geometry of local minima and saddle points for the renormalized Cahn–Hilliard energy $\mathcal{E}_{\phi}^{\xi}(u)$ on the torus in the critical regime with $L\sim\phi^{-(d+1)/d}$ and $\phi\ll 1$. It combines a numerical investigation in $d=2$ using the String Method to locate a minimum and an intermediate saddle with a droplet-like shape, and a theoretical convexity analysis of level sets via a Caffarelli–Spruck approach applied to the Euler–Lagrange equation $\Delta u=f(u)$. The authors establish that, under suitable hypotheses, superlevel sets are convex up to the maximum value and provide detailed optimality conditions and contradiction arguments for cases $0<t_0<1$ and $t_0=1$ to support this convexity claim. These results illuminate nucleation-type transitions in CH energy and the geometry of critical points, connecting numerical observations with rigorous geometric properties of minimizers and saddles. The work advances understanding of droplet-like structures, Steiner symmetry, and the convexity of level sets in CH-type energies, with implications for phase separation and nucleation theory.

Abstract

We explore recent progress and open questions concerning local minima and saddle points of the Cahn--Hilliard energy in $d\geq 2$ and the critical parameter regime of large system size and mean value close to $-1$. We employ the String Method of E, Ren, and Vanden-Eijnden -- a numerical algorithm for computing transition pathways in complex systems -- in $d=2$ to gain additional insight into the properties of the minima and saddle point. Motivated by the numerical observations, we adapt a method of Caffarelli and Spruck to study convexity of level sets in $d\geq 2$.

Numerics and analysis of Cahn--Hilliard critical points

TL;DR

The paper addresses the existence and geometry of local minima and saddle points for the renormalized Cahn–Hilliard energy on the torus in the critical regime with and . It combines a numerical investigation in using the String Method to locate a minimum and an intermediate saddle with a droplet-like shape, and a theoretical convexity analysis of level sets via a Caffarelli–Spruck approach applied to the Euler–Lagrange equation . The authors establish that, under suitable hypotheses, superlevel sets are convex up to the maximum value and provide detailed optimality conditions and contradiction arguments for cases and to support this convexity claim. These results illuminate nucleation-type transitions in CH energy and the geometry of critical points, connecting numerical observations with rigorous geometric properties of minimizers and saddles. The work advances understanding of droplet-like structures, Steiner symmetry, and the convexity of level sets in CH-type energies, with implications for phase separation and nucleation theory.

Abstract

We explore recent progress and open questions concerning local minima and saddle points of the Cahn--Hilliard energy in and the critical parameter regime of large system size and mean value close to . We employ the String Method of E, Ren, and Vanden-Eijnden -- a numerical algorithm for computing transition pathways in complex systems -- in to gain additional insight into the properties of the minima and saddle point. Motivated by the numerical observations, we adapt a method of Caffarelli and Spruck to study convexity of level sets in .

Paper Structure

This paper contains 14 sections, 15 theorems, 148 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background and open questions
  3. Quantitative bounds
  4. Numerical results
  5. The String Method
  6. Implementation and convergence
  7. Numerical results
  8. Convexity
  9. Convexity method of Caffarelli and Spruck
  10. Optimality conditions for extremals
  11. The case $0<t_0<1$
  12. Contradiction to the optimality of $(x_0,y_0,z_0)$ in the case $t_0<1$.
  13. The case $t_0=1$
  14. Contradiction to the optimality of $(x_0,y_0,z_0)$ in the case $t_0=1$.

Key Result

Lemma 4.1

For any $\omega_1>0$ there exists $\phi_0>0$ such that for all $\phi\in(0,\phi_0)$, any volume-constrained minimizer $u$ for $\omega\in[\omega_1,\xi^{3}/2]$ satisfies

Figures (4)

  • Figure 2.1: The function $f_\xi$ defined in \ref{['f']} has for $\tilde{\xi}_d<\xi<\xi_d$ a local (but not global) minimum at $\nu_m>0$ (left figure) and for $\xi>\xi_d$ a global minimum at $\nu_m>0$ (right figure). For $\xi<\tilde{\xi}_d$, the only local minimum is at zero (not shown).
  • Figure 3.1: The local minimizer has the expected properties. Left: Level curves of the local minimum for $\phi = 0.1, \xi = 1.9$. Right: Crossections at various $y$ as a function of $x$.
  • Figure 3.2: Local minimizer (left) and saddle point (right) for $\phi=0.1$ and $\xi=1.6$, i.e., in the regime that the constant state is not the global minimizer. The string to compute the saddle point is converged up to error $10^{-5}$.
  • Figure 3.3: A close-up of the level lines of the saddle point for $\phi=0.2$ and $\xi=1.9$. The string is converged up to error $10^{-4}$. This figure motivates the analysis in Section \ref{['S:convex']}.

Theorems & Definitions (36)

  • Remark 1.1
  • Remark 3.1: Generalizations
  • Lemma 4.1
  • Theorem 4.2: Theorem 1.1, GWW20
  • Remark 4.3
  • Remark 4.4
  • Lemma 4.5
  • proof
  • Theorem 4.6
  • Definition 4.7
  • ...and 26 more