[Paper Review] Uniqueness results for critical points of a non-local isoperimetric problem via curve shortening

Dorian Goldman|arXiv (Cornell University)|Jun 26, 2012

Advanced Mathematical Modeling in Engineering44 references3 citations

TL;DR

This paper establishes the uniqueness of connected critical points for a non-local isoperimetric problem arising in di-block copolymer melts using area-preserving curve shortening flow and a novel curvature-potential inequality. It proves that in ℝ², under small mass/energy conditions, the only connected critical point is the ball, and on the torus, only the ball and stripe pattern are possible, extending classical isoperimetric and curve shortening results to non-local settings.

ABSTRACT

Using area-preserving curve shortening flow, and a new inequality relating the potential generated by a set to its curvature, we study a non-local isoperimetric problem which arises in the study of di-block copolymer melts, also referred to as the Ohta-Kawasaki energy. We are able to show that the only connected critical point is the ball under mild assumptions on the boundary, in the small energy/mass regime. In particular this class includes all rectifiable, connected 1-manifolds in $\mathbb{R}^2$. We also classify the simply connected critical points on the torus in this regime, showing the only possibilities are the stripe pattern and the ball. In $\mathbb{R}^2$, this can be seen as a partial union of the well known result of Fraenkel \cite{Fraenkel} for uniqueness of critical points to the Newtonian Potential energy, and Alexandrov for the perimeter functional \cite{alexandrov}, however restricted to the plane. The proof of the result in $\mathbb{R}^2$ is analogous to the curve shortening result due to Gage \cite{Gage2}, but involving a non-local perimeter functional, as we show the energy of convex sets strictly decreases along the flow. Using the same techniques we obtain a stability result for minimizers in $\mathbb{R}^2$ and for the stripe pattern on the torus, the latter of which was recently shown to be the global minimizer to the energy when the non-locality is sufficiently small \cite{sternberg}.

Motivation & Objective

• To establish uniqueness of connected critical points for the Ohta-Kawasaki energy in ℝ² and on the torus under small mass/energy regimes.
• To extend classical isoperimetric and curve shortening results to non-local perimeter functionals arising in di-block copolymer systems.
• To prove that the ball is the unique simply connected critical point in ℝ² and classify critical points on the torus in the small energy regime.
• To provide a stability result for minimizers in ℝ² and for the stripe pattern on the torus, confirming their optimality under small non-locality.

Proposed method

• Uses area-preserving curve shortening flow to evolve sets and show energy decreases for convex sets, implying non-constant curvature sets cannot be critical points.
• Introduces a new inequality linking the potential generated by a set to its curvature, essential for controlling the non-local term.
• Applies the first variation of the energy along volume-preserving diffeomorphisms to characterize critical points via the Euler-Lagrange equation involving curvature and potential.
• Employs rescaling techniques to reduce the general Ohta-Kawasaki energy to a form with unit non-local coefficient, simplifying analysis.
• Uses the regularity result from Sternberg and Topaloglu (2023) on C³,α boundary regularity of minimizers to justify the flow and energy decay estimates.
• Constructs explicit counterexamples (annuli) to show that non-constant curvature sets can satisfy the Euler-Lagrange equation under specific kernel choices, highlighting the necessity of the small mass assumption.

Experimental results

Research questions

• RQ1 What are the unique critical points of the non-local isoperimetric problem in ℝ² under small mass conditions?
• RQ2 Can the curve shortening flow technique be adapted to non-local perimeter functionals to prove uniqueness of critical points?
• RQ3 What are the simply connected critical points on the torus for the Ohta-Kawasaki energy in the small non-locality regime?
• RQ4 Is the ball the only stable minimizer in ℝ² for small masses, and does the stripe pattern remain stable on the torus?
• RQ5 How does the interplay between curvature and non-local potential affect the existence and uniqueness of critical configurations?

Key findings

• In ℝ², under mild boundary regularity and small mass/energy conditions, the only connected critical point is the ball, extending Fraenkel’s and Alexandrov’s results to the non-local setting.
• On the torus, the only simply connected critical points in the small energy regime are the ball and the stripe pattern with n=1, confirming a conjecture on phase separation patterns.
• The energy of convex sets strictly decreases along the area-preserving curve shortening flow, implying that only sets with constant curvature can be critical points.
• A stability result is established: minimizers in ℝ² and the stripe pattern on the torus are stable under small perturbations when the non-local parameter is sufficiently small.
• Explicit counterexamples (annuli with specific radii) are constructed where non-constant curvature sets satisfy the Euler-Lagrange equation, showing that the small mass assumption is necessary for uniqueness.
• The paper proves that the first variation of energy vanishes at t=0 only if curvature is constant, which characterizes the ball and stripe as the only possible critical points in the regimes studied.