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First Dirichlet eigenvalue of the weighted 1-Laplacian operator

This paper establishes that the first Dirichlet eigenvalue of the weighted 1-Laplacian operator coincides with the weighted Cheeger constant by analyzing the limit of the weighted $p$ -Laplacian eigenvalues as $p \to 1$ and proving the convergence of the corresponding eigenfunctions.

Original authors: R. Barbato, J. C. Sabina de Lis, S. Segura de León

Published 2026-05-26

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CC BY 4.0

Original authors: R. Barbato, J. C. Sabina de Lis, S. Segura de León

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 rubber sheet stretched over a frame (a shape called $\Omega$ ). In mathematics, we often want to know how this sheet vibrates or settles. The "eigenvalue" is like the lowest possible pitch the sheet can make when it vibrates. The lower the pitch, the easier it is for the sheet to move.

This paper is about a very specific, tricky kind of rubber sheet where the material isn't the same everywhere. Some parts are thick and heavy (weighted by a function $a(x)$ ), and some parts are light. Also, the way the sheet resists being pulled isn't the usual "springy" resistance; it's a very stiff, almost rigid resistance (the "1-Laplacian").

Here is the story of what the authors discovered, broken down into simple concepts:

1. The Two Ways to Measure "Stiffness"

The authors were trying to connect two different ways of measuring how hard it is to move this weighted sheet.

Method A (The Vibration Test): This is the "Eigenvalue." It asks: "What is the lowest energy state this sheet can have?" You calculate this by looking at how the sheet vibrates mathematically.
Method B (The Cheese Cut): This is the "Cheeger Constant." Imagine you want to cut a piece of cheese (the shape) into two pieces with the least amount of cutting effort. You want to find a cut that separates a small piece of cheese from the rest, but the cut itself is as short as possible. The "Cheeger constant" is a ratio: How much "cutting edge" do you need per unit of "cheese volume"?

In the world of standard, uniform rubber sheets, mathematicians already knew these two numbers were the same. But this paper asks: Does this still hold true when the sheet is weighted (uneven) and the physics is weird (the 1-Laplacian)?

2. The "Smooth" Approach to a "Rough" Problem

The "1-Laplacian" is a mathematical monster. It's so sharp and singular that it's hard to work with directly. It's like trying to measure the slope of a cliff with a ruler; the ruler just doesn't fit.

To solve this, the authors used a clever trick called asymptotic analysis:

They started with a "smooth" version of the problem (the $p$ -Laplacian), where $p$ is a number slightly larger than 1. Think of this as a slightly flexible rubber sheet that is easy to measure.
They calculated the "lowest pitch" (eigenvalue) for this smooth sheet.
Then, they slowly turned the knob, making $p$ get closer and closer to 1. They watched what happened to the pitch as the sheet became more and more rigid.

3. The Big Discovery

The authors proved that as the sheet becomes perfectly rigid ( $p \to 1$ ):

The "lowest pitch" (the eigenvalue) settles down to a specific number.
That specific number is exactly the same as the "Cheeger constant" (the most efficient cut).

In other words, the hardest way to vibrate the sheet is mathematically identical to the most efficient way to cut it.

4. The "Weight" Matters

The paper deals with "weights." Imagine the sheet has patches of lead sewn onto it ( $a(x)$ ) and the cheese has patches of honey ( $b(x)$ ).

The authors showed that even with these heavy, uneven patches, the rule still holds.
They had to be careful about how "rough" or "smooth" these weights were. If the weights were too jagged (not Lipschitz continuous), the math gets messy. However, they proved that even for very general, rough weights, the rule holds: the limit of the vibrations equals the Cheeger constant.

5. The Shape of the Solution

When you find the "lowest pitch" for this rigid sheet, what does the sheet look like?

For the smooth sheet, the shape is a gentle curve.
For the rigid sheet (the limit), the shape turns out to be a step function. It looks like a flat plateau that suddenly drops off.
The authors proved that this "flat plateau" shape is actually a "Cheeger set"—it's the exact shape you would get if you made the most efficient cut through the weighted domain.

Summary Analogy

Imagine you are a city planner trying to build a wall to separate a city into two districts.

The Cheeger Constant asks: "What is the shortest wall I can build to separate a specific amount of population?"
The Eigenvalue asks: "If I blow a whistle, what is the lowest note the city's layout will naturally resonate at?"

This paper proves that for a city with uneven terrain (weights) and rigid zoning laws (1-Laplacian), the shortest wall you can build is exactly the same number as the lowest note the city will hum. The geometry of the cut and the physics of the vibration are two sides of the same coin.

Technical Summary: First Dirichlet Eigenvalue of the Weighted 1-Laplacian Operator

Problem Statement
This paper investigates the spectral properties of the weighted 1-Laplacian operator under Dirichlet boundary conditions. Specifically, the authors analyze the eigenvalue problem:
$\begin{cases} -\text{div}\left( a(x) \frac{Du}{|Du|} \right) = \Lambda b(x) \frac{u}{|u|} & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \end{cases}$
where $\Omega \subset \mathbb{R}^N$ ( $N \ge 2$ ) is a bounded domain with Lipschitz boundary, and $a(x)$ and $b(x)$ are non-negative weight functions. The primary objective is to establish that the first eigenvalue $\Lambda$ of this problem coincides with the weighted Cheeger constant of the domain. Furthermore, the paper addresses the asymptotic behavior of the first eigenvalue $\lambda_p$ of the corresponding weighted $p$ -Laplacian problem as $p \to 1$ , aiming to prove the convergence of both eigenvalues and eigenfunctions to the limit problem.

Methodology
The authors employ a limiting procedure, analyzing the sequence of weighted $p$ -Laplacian eigenvalue problems as $p \to 1^+$ . The methodology relies on the following technical framework:

Weighted Functional Spaces: The study is conducted within weighted Sobolev spaces $W^{1,p}_0(\Omega, a)$ and weighted spaces of functions of bounded variation $BV(\Omega, a)$ . The weights $a(x)$ and $b(x)$ satisfy specific integrability conditions: $a \in W^{1,1}(\Omega)$ with $a(x) \ge \mu > 0$ , and $b \in L^r(\Omega)$ with $r > N$ .
Anzellotti's Pairing Theory: To handle the singular nature of the 1-Laplacian (where the gradient is a Radon measure), the authors extend Anzellotti's theory of pairing between vector fields and gradients of BV functions. They establish a weighted Green's formula involving the pairing $(z, Du)$ and the trace of the normal component $[z, \nu]$ , allowing for the treatment of unbounded weights and boundary terms without resorting to abstract distributions for every term.
Compactness and Convergence: The existence of the limit is derived by establishing uniform bounds on the sequence of $p$ -eigenfunctions. Using the compact embedding of $BV(\Omega, a)$ into $L^q(\Omega)$ , the authors prove that the sequence of normalized eigenfunctions converges strongly to a limit function $u$ .
Variational Characterization: The authors utilize the co-area formula and properties of the weighted perimeter to relate the variational formulation of the 1-Laplacian to the weighted Cheeger constant.

Key Contributions and Results

Identification of the First Eigenvalue: The paper proves that the first eigenvalue of the weighted 1-Laplacian is exactly the weighted Cheeger constant, defined as:
$h_p(\Omega, a, b) = \inf_{E} \frac{P_a(E, \mathbb{R}^N)}{\int_E b(x) dx},$
where the infimum is taken over measurable subsets $E \subset \Omega$ with finite weighted perimeter, and $P_a$ denotes the weighted perimeter.
Convergence of Eigenvalues:
- For Lipschitz continuous weights $a(x)$ , the authors prove that the limit $\lim_{p \to 1} \lambda_p(\Omega, a, b)$ exists and equals the weighted Cheeger constant. In this case, the "smooth" weighted Cheeger constant (defined via smooth domains) coincides with the general weighted Cheeger constant.
- For general weights (where $a \in W^{1,1}(\Omega)$ ), the paper establishes that the least eigenvalue of the limit problem is the weighted Cheeger constant, even if the existence of the limit of $\lambda_p$ is not explicitly proven for all sequences.
Boundedness of Eigenfunctions: A significant result is the proof that the eigenfunction associated with the first eigenvalue of the weighted 1-Laplacian is bounded ( $u \in L^\infty(\Omega)$ ). This is achieved through a Moser-type iteration argument adapted to the weighted setting, utilizing the condition $b \in L^r(\Omega)$ with $r > N$ .
Convergence of Eigenfunctions: The normalized sequence of $p$ -eigenfunctions converges (up to a subsequence) to the normalized eigenfunction of the limit problem. The limit function satisfies the definition of an eigenpair involving a vector field $z \in L^\infty(\Omega, \mathbb{R}^N)$ with $\|z\|_\infty \le 1$ and a function $\gamma \in \text{sign}(u)$ .

Significance and Claims
The paper claims to provide a rigorous foundation for the spectral theory of the weighted 1-Laplacian, a problem that had not been fully addressed in the literature prior to this work. By identifying the first eigenvalue with the weighted Cheeger constant, the authors bridge the gap between spectral theory and geometric optimization (isoperimetric inequalities) in the context of weighted operators.

The authors emphasize that their work extends previous results on the unweighted 1-Laplacian (where the first eigenvalue is the standard Cheeger constant) to a general weighted setting. They explicitly note that their approach avoids "spurious solutions" often encountered in non-smooth critical point theory by constructing the solution as a limit of $p$ -Laplacian eigenfunctions. The proof of the boundedness of the eigenfunction is highlighted as a crucial step that ensures the solution is well-defined and allows for the application of the Green's formula in the limit.

The paper concludes that for Lipschitz weights, the asymptotic behavior of the $p$ -Laplacian spectrum is well-behaved, converging to the geometrically defined Cheeger constant, thereby validating the use of the $p \to 1$ limit as a robust method for studying the 1-Laplacian spectrum.