🔢 mathematics

On the second anisotropic Cheeger constant and related questions

This paper investigates the asymptotic behavior of the second eigenfunction of the anisotropic $p$ -Laplace operator as $p \to 1^+$ , introduces the second anisotropic Cheeger constant to establish its connection with the second eigenvalue, and analyzes the twisted anisotropic $q$ -Cheeger constant under a volume constraint.

Original authors: Gianpaolo Piscitelli

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Gianpaolo Piscitelli

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a piece of clay (a shape called $\Omega$ ) and you want to understand its "vibrations." In mathematics, these vibrations are described by something called eigenvalues. Think of the first eigenvalue as the lowest, deepest hum the clay can make, and the second eigenvalue as the next higher pitch.

This paper is a mathematical investigation into what happens to these vibrations when we change the rules of the game slightly, specifically as a parameter $p$ gets closer and closer to 1.

Here is a breakdown of the paper's main ideas using simple analogies:

1. The "Anisotropic" Twist: The Shape of the Clay

Usually, math problems assume space is the same in every direction (like a perfect sphere). This paper deals with anisotropic problems.

The Analogy: Imagine your clay isn't a sphere, but a weirdly shaped crystal or a stretched-out blob. In this world, moving "North" might be easier or harder than moving "East." The paper uses a special ruler (called a norm $F$ ) to measure distances and shapes that respects this unevenness.
The Goal: The author wants to see how the "vibrations" of this weirdly shaped clay behave when the rules of the game change.

2. The "Cheeger" Problem: Cutting the Cake

To understand the vibrations, the author looks at a concept called the Cheeger constant.

The Analogy: Imagine you have a cake (your shape). You want to cut it into two pieces with a single slice.
- The First Cheeger Constant asks: "What is the most efficient way to cut off a single piece of cake so that the slice (the perimeter) is as small as possible compared to the size of the piece?" It's like finding the easiest way to take a bite out of the cake without wasting too much crust.
- The Second Cheeger Constant (the main star of this paper) asks: "What is the most efficient way to cut the cake into two separate pieces at the same time?" You want to find two chunks of cake where the total amount of "crust" (perimeter) needed to separate them from the rest is minimized relative to their size.

3. The Big Discovery: Vibrations Meet Cutting

The paper proves a fascinating connection between the vibrations (eigenvalues) and the cutting (Cheeger constants) as the parameter $p$ approaches 1.

The First Result (Old News, but confirmed): It was already known that the lowest vibration (first eigenvalue) eventually becomes exactly equal to the efficiency of the single best cut (first Cheeger constant).
The New Result (The Paper's Contribution): The author proves that the second vibration (the second eigenvalue) also settles down to match the second Cheeger constant.
- In plain English: As the rules of the game get closer to the limit, the "pitch" of the second vibration becomes exactly equal to the "efficiency score" of the best way to split the shape into two pieces.

4. The "Twisted" Constant: A Volume Constraint

The paper also looks at a slightly different problem called the "twisted" Cheeger constant.

The Analogy: Imagine you are cutting the cake, but you have a strict rule: the two pieces you cut out must balance each other perfectly. If one piece is heavy (positive), the other must be equally heavy (negative) so the total weight is zero.
The author shows that this "balanced cut" problem sits right in the middle of the first and second Cheeger constants. It's a bridge between cutting off one piece and splitting the whole thing into two.

5. The "Perfect" Shapes

Finally, the paper asks: "What shape of cake minimizes these cutting costs?"

The Answer: The most efficient shapes are Wulff shapes.
- The Analogy: If your ruler is a perfect circle, the Wulff shape is a circle. If your ruler is a square, the Wulff shape is a square. These are the "perfect" shapes for this specific type of geometry.
The author proves that when you try to minimize the "twisted" cost, the best shape is always a combination of two of these perfect Wulff shapes sitting side-by-side. Interestingly, depending on the specific math settings, these two shapes might be the same size or different sizes.

Summary

In short, this paper is a rigorous mathematical proof showing that for a specific type of uneven geometry:

The second vibration of a shape eventually becomes identical to the best way to split that shape into two pieces.
The most efficient shapes for these problems are always combinations of the geometry's "perfect" building blocks (Wulff shapes).

The author uses advanced tools (like "BV functions," which are like mathematical descriptions of shapes with sharp edges) to prove that even though the math is complex, the relationship between the "sound" of the shape and the "cut" of the shape is direct and predictable.

Technical Summary: On the Second Anisotropic Cheeger Constant and Related Questions

Problem Statement
This paper investigates the asymptotic behavior of the second eigenvalue, $\lambda_{2,F}(p, \Omega)$ , of the anisotropic $p$ -Laplace operator as the exponent $p$ approaches 1 from above ( $p \to 1^+$ ). The operator is defined as $-Q_p u := -\text{div}(F^{p-1}(\nabla u)\nabla_\xi F(\nabla u))$ , where $F$ is a smooth, convex, 1-homogeneous norm on $\mathbb{R}^n$ . While the relationship between the first eigenvalue and the first anisotropic Cheeger constant ( $h_{1,F}$ ) is well-established in the limit $p \to 1^+$ , the behavior of the second eigenvalue and its connection to a corresponding "second" Cheeger constant ( $h_{2,F}$ ) requires further rigorous analysis. The paper also explores a constrained minimization problem related to a "twisted" anisotropic $q$ -Cheeger constant.

Methodology
The author employs variational methods and tools from Finsler geometry and the theory of functions of bounded variation (BV). Key methodological components include:

Variational Characterization: Utilizing the min-max characterization of the second eigenvalue via the Lusternik-Schnirelmann theory and the decomposition of the domain into two disjoint subdomains.
Approximation Techniques: Leveraging interior approximation results for BV functions by smooth functions with compact support (specifically citing [BDPP, Theorem 4.3]) to handle regularity issues without imposing restrictive assumptions on the domain boundary.
Coarea Formulas: Applying anisotropic coarea formulas to relate the total variation of functions to the anisotropic perimeter of their level sets.
Isoperimetric Inequalities: Using the anisotropic isoperimetric inequality, which states that the Wulff shape minimizes the anisotropic perimeter for a fixed volume, to derive lower bounds and characterize minimizers.
Twisted Constant Analysis: Reformulating the twisted Cheeger constant problem as a minimization over disjoint sets and reducing it to a one-dimensional optimization problem involving Wulff shapes.

Key Contributions and Results

Anisotropic Cheeger Inequality for the Second Eigenvalue:
The paper establishes a Cheeger-type inequality for the second eigenvalue:
$\lambda_{2,F}(p, \Omega) \geq \left( \frac{h_{2,F}(\Omega)}{p} \right)^p$
where the second anisotropic Cheeger constant is defined as:
$h_{2,F}(\Omega) := \inf \left\{ \max \left\{ \frac{P_F(E_1)}{|E_1|}, \frac{P_F(E_2)}{|E_2|} \right\} : E_1, E_2 \subset \Omega, E_1 \cap E_2 = \emptyset, |E_1|, |E_2| > 0 \right\}$
Here, $P_F(E)$ denotes the anisotropic perimeter.
Limiting Behavior as $p \to 1^+$ :
The primary result proves that the second eigenvalue converges to the second Cheeger constant in the limit:
$\lim_{p \to 1^+} \lambda_{2,F}(p, \Omega)^{1/p} = h_{2,F}(\Omega)$
The proof involves showing both the lower bound (via the Cheeger inequality) and the upper bound (by constructing test functions based on the minimizing sets for $h_{2,F}$ ).
Convergence of Eigenfunctions:
The paper demonstrates that the normalized second eigenfunctions $u_{2,p}$ converge in $L^1(\Omega)$ (up to a subsequence) to a function $u_2 \in BV(\Omega)$ with unit $L^1$ norm. This limit function satisfies $\int_\Omega |Du_2|_F \leq h_{2,F}(\Omega)$ and is not strictly positive or negative, reflecting the nodal structure of the second eigenfunction.
Discrepancy for Higher Eigenvalues:
The author notes that while the limit holds for $k=2$ , for $k \geq 3$ , only the inequality $\limsup_{p \to 1^+} \lambda_{k,F}(p, \Omega)^{1/p} \leq h_{k,F}(\Omega)$ can be guaranteed. This discrepancy arises because the $k$ -th eigenfunction typically does not possess $k$ distinct nodal domains.
Twisted Anisotropic $q$ -Cheeger Constant:
The paper studies the constant $K_{q,F}(\Omega)$ , defined as the infimum of the ratio of the anisotropic total variation to the $L^q$ norm of functions with zero mean. It is shown that $h_{1,F}(\Omega) \leq K_{1,F}(\Omega) \leq h_{2,F}(\Omega)$ .
- Minimization: The minimizer of $K_{q,F}(\Omega)$ over sets of fixed volume is proven to be the union of two disjoint Wulff shapes.
- Uniqueness and Geometry: A critical exponent $\tilde{q}(n)$ is identified. For $q < \tilde{q}$ , the unique minimizer consists of two disjoint Wulff shapes of equal radius. For $q > \tilde{q}$ , the unique minimizer consists of two disjoint Wulff shapes of different radii. For $n=2$ , $\tilde{q} = 7/4$ , and the minimizer remains unique and symmetric even at the critical value.

Significance and Claims
The paper claims to extend the known asymptotic theory of the $p$ -Laplacian from the first eigenvalue to the second eigenvalue in the anisotropic setting. By proving the convergence of $\lambda_{2,F}$ to $h_{2,F}$ , the work bridges the gap between spectral theory and geometric optimization for the second mode. The author explicitly states that the results overcome certain regularity assumptions found in previous literature (such as those in [Li, S]) by utilizing interior approximation techniques for BV functions. Furthermore, the characterization of the twisted Cheeger constant minimizers provides a geometric understanding of the constraints imposed by the zero-mean condition in the anisotropic context, generalizing previous Euclidean results. The paper does not propose new experimental applications but rather solidifies the theoretical framework connecting nonlinear eigenvalue problems, anisotropic geometry, and Cheeger-type constants.