Regularity of Crystalline Almost-Minimisers in the Plane
Speaker: Eric KimDate of Talk: April 23, 2025
Upstream link: Participating Analysis Seminar
1. The Setup
Imagine a drop of water, possibly falling through space or otherwise under the influence of some potential. The droplet will seek to minimise its energy, which comes from both its surface tension and this external potential. Specifically, we are seeking a way to minimise \[\underbrace{\int _{\partial E} \phi \left( \nu _E \right) d \mathcal{H} ^{n-1}}_{P_ \phi (E)} + \int _{E} g\ dx\] over all sets \(E\) of fixed volume \(m\). Here, \(\phi \) describes the energy density on the surface of \(E\) and \(g\) is the potential acting on \(E\).
Rather than finding a bona fide minimiser of this energy, one instead seeks almost-minisers:
Definition 1.
\(E\) is a \(\left( \Lambda , r_0 \right)\)- minimiser if for all \(x\) and \(G\) with \(E \Delta G \subset\subset B \left( x, r_0 \right)\), one has \[P_ \phi (E) \leq P_ \phi (G) + \Lambda \left\lvert E \Delta G \right\rvert.\]
If \(\phi \) is smooth and elliptic, then it’s known that almost-minimsers are \(C ^{1, \alpha }\) away from a singular set \(S\) with \(\mathcal{H} ^{n-3} (S) = 0\).
What happens for other \(\phi \)? In practise, a lot of people care about “crystalline” choices of \(\phi \), i.e. piecewise linear energy densities. Among other things, Figalli and Maggi in 2011 proved that
Theorem 2.
When \(n=2\), \(\Lambda \ll 1\), and \(r_0 \\gamma 1\), \(\left( \Lambda ,r_0 \right)\)-minimisers are convex polygons with sides parallel to the Wulff shape of \(\phi \).
2. The Main Result
The speaker and his coauthors proved that in \(n=2\), assuming \(\Lambda r_0 \leq 1\) and \(\phi \) is crystalline, then \(\left( \Lambda , r_0 \right)\)-minimisers are “Lipschitz \(\phi \)-regular”. Additionally, for some constant \(c > 0\) depending only on \(\phi \), \(\partial E \cap B \left( x, cr_0 \right)\) is a “\(\phi \)-geodesic”.
Somewhere along the way, one also needs the assumption that there are at least \(6\) normal vectors to the Wulff shape.
One starts with an almost-minimiser \(E\). Tools from geometric measure theory tell us that \(\partial E\) is comprised of finitely many rectifiable curves, and one can then show that each of these curves are Lipschitz \(\phi \)-regular and that, at small scales, they are \(\phi \)-geodesics. This is done by just wiggling the boundary of \(E\) at small scales and replacing it with the “right thing”. Thus any energy-saving perturbations must happen at very large scales.
When these curves join together, the overlapping Lipschitz \(\phi \)-regularity conditions force a “corner” to form, or a facet.
3. \(n \geq 3\)?
When \(n \geq 3\), one of the barriers to is that the geometric-measure-theory-black-box of writing \(\partial E\) as rectifiable curves is no longer accessible. Additionally, the idea of a “\(\phi \)-geodesic” becomes hazy in higher dimensions — one needs to encode some data about boundaries on a ball. Eric said he has “no hope” of extending this result in that direction.