E. Paolini: Infinite Isoperimetric Partitions (Trento 2025)

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infinite
isoperimetric
partitions

[E. Paolini: Università di Pisa]

Trento ‐ Jan 28, 2025

paolini.github.io/2025-talk-Trento

Plan

infinite isoperimetric clusters
[Novaga - P. - Stepanov - Tortorelli]
locally isoperimetric partitions
[Novaga - P. - Tortorelli]
periodic isoperimetric tilings
[Nobili - Novaga - P.]

Part 1.

infinite isoperimetric clusters

minimize $\displaystyle P(E)= \frac 1 2 \sum_{k=0}^{+\infty} P(E_k)$

with fixed $\lvert E_k \rvert$, $\displaystyle\sum_{k=1}^{+\infty} \lvert E_k\vert \lt +\infty$

Concentration compactness?

[Novaga - Paolini - Stepanov - Tortorelli
J. Geom. Anal. 2022]

boundary of an infinite cluster

$$\begin{aligned} \mathbf{E} &= (E_1,\dots,E_k,\dots), \qquad E_k\subset \mathbb{R}^d \\ \partial^* \mathbf E &= \bigcup_{k=1}^{+\infty} \bigcup_{j=0}^{k-1}\partial^* E_k \cap \partial^* E_j \\ \partial \mathbf E &= \left\{ x\mid \forall \rho\, \exists k\colon 0 \lt \lvert E_k\cap B_\rho(x)\rvert \lt \lvert B_\rho(x)\rvert \right\}\\ \end{aligned}$$

Lemma: $P(\mathbf E) = \mathcal{H}^{d-1}(\partial^* \mathbf E)$

Theorem. Given $a_k\gt 0$, $k=1,2,\dots$ such that $\sum \sqrt {a_k} \lt \infty$, there exists a minimal cluster $\mathbf E$ in $\mathbb R^2$ with $\lvert E_k\rvert = a_k$. Moreover $\mathcal H^1(\partial \mathbf E \setminus \partial^* \mathbf E) = 0$.

[Novaga - P. - Stepanov - Tortorelli
networks and heterogeneous media 2023]

proof. $\mathbf E^n$ minimizing for $a_1,\dots,a_n$.
Uniform estimate on the diameter of the clusters.
$E_k^n \to E_k$, $\quad\mathbf E = (E_1,E_2\dots)$.
Blaschke: $\partial \mathbf E^n \to K$.
Goła̧b: $\mathcal H^1(K)\le \lim \mathcal H^1(\partial \mathbf E^n)$. Finally $ P(\mathbf E) = \mathcal H^1(\partial^* \mathbf E) \le \mathcal H^1(\partial \mathbf E) \le \mathcal H^1(K) \le \lim P(\mathbf E^n) \le \lim P(\mathbf E) $.

Remark. $\mathcal H^1(\partial \mathbf E \setminus \partial^* \mathbf E) = 0$ implies $P(\mathbf E) = \mathcal H^1(\partial \mathbf E)$ but does not imply $P(E_k) = \mathcal H^1(\partial E_k)$ for all $k$. Not even in a finite cluster:

[Ambrosio - Caselles - Masnou - Morel
JEMS 2001]

Part 2.

locally isoperimetric partitions

$\mathbf E = (E_1,\dots,E_N)$ is a partition of $\mathbb R^d$. Find $\mathbf E$ minimizing $P(\mathbf E, \Omega)$ on every bounded $\Omega$ with $\lvert E_k \rvert$ fixed, but possibly $\infty$.

[Alama - Bronsard - Vriend
to appear on: Transactions of the AMS ]

planar standard bubbles
$\to$ standard partitions

$\to$
$\to$ $\to$

[Foisy et al.
Pacific J. Math. 1993] [Wichiramala
J. Reine Angew. Math 2004]

Higer dimensions

$\to$
$\to$ $\to$

[Hutchings - Morgan - Ritoré
Annals of Math. 2002] [Reichardt
J. Geom. Anal. 2008] [Milman - Neeman
to appear: acta mathematica]

$J$-isoperimetric partitions

$\mathbf E$ minimizes $P(\mathbf E, \Omega)$ with fixed $\lvert E_j\cap \Omega\rvert$ for all $j\in J \subset (1,2,\dots, N)$.

Theorem. $J$-isoperimetric partitions are closed with respect to $L^1_{\text{loc}}$ convergence.

Theorem. $J$-isoperimetric partitions are locally isoperimetric if their boundary is flat at infinity
($d\le 7$)

[Novaga - P. - Tortorelli
to appear on: Transactions of the AMS] [Bronsard - Novack
arXiv 2024]

different notions of minimality

Locally isoperimetric partitions allow the exchange of any volume between infinite regions whereas
$J$-isoperimetric partitions don't.
👉 Large exchange of volumes is required to pass from one notion to the other.

future plans

Show that in $\mathbb R^8$ there exists a locally isoperimetric partition with one finite region and two infinite regions which coincides with the Simons' cone at infinity.

Part 3.

periodic isoperimetric double-tilings

$G$ discrete subgroup of $\mathbb R^2$ (lattice).
$\displaystyle \mathbf E = \bigcup_{g\in G}\{E_1+g,E_2+g\} $ partition of $\mathbb R^d$. minimize $P(E_1)+P(E_2)$ with $\lvert E_1 \rvert$ and $\lvert E_2\rvert$ fixed.

[Nobili - Novaga - P.
draft]