Pleijel Theorem in Spectral Geometry

• Pleijel Theorem is a key result in spectral geometry that links the asymptotic behavior of nodal domains of Laplacian eigenfunctions to geometric inequalities.
• The theorem refines Courant’s bound by utilizing the Faber–Krahn inequality and Weyl’s law, demonstrating that equality holds only for finitely many eigenpairs in dimensions two and higher.
• Extensions of the theorem cover Neumann, Robin, Schrödinger, and sub-Riemannian operators, offering practical insights into geometric optimization and spectral partition problems.

The Pleijel theorem is a foundational result in spectral geometry and analysis concerning the asymptotic behavior of the number of nodal domains of Laplacian eigenfunctions, asserting that equality in Courant's nodal domain bound occurs only for finitely many eigenpairs in dimension at least two. Since Pleijel’s original 1956 argument, the theorem has become central in understanding the geometric and topological features of eigenfunctions in numerous settings, including Riemannian, sub-Riemannian, and non-smooth spaces, and for generalizations to a variety of operators and boundary conditions. The theorem quantitatively refines Courant’s theorem and forms the basis for much of the asymptotic spectral theory concerning nodal geometry, spectral minimal partitions, and quantitative isoperimetric analysis.

1. Classical Formulation and Proof Outline

Given a bounded domain $\Omega \subset \mathbb{R}^d$ (with $d\geq 2$), consider the eigenvalue problem for the Dirichlet Laplacian: $$-\Delta u_n = \lambda_n u_n \quad \text{in } \Omega, \qquad u_n|_{\partial \Omega} = 0,$$ where $$0 < \lambda_1 < \lambda_2 \leq \lambda_3 \leq \cdots$$ are arranged in non-decreasing order including multiplicities. Courant’s nodal domain theorem asserts that the number $\mu(u_n)$ of nodal domains of $u_n$ satisfies $\mu(u_n)\leq n$. Pleijel’s theorem (Bourgain, 2013) sharpens this by establishing that

$$\limsup_{n\to\infty}\frac{\mu(u_n)}{n} \leq \frac{4}{j^2} < 1,$$

where $j\approx 2.4048$ is the first positive zero of $J_0$ (the Bessel function of the first kind), with $j^2$ the first Dirichlet eigenvalue of the unit disk. Thus, except for finitely many eigenfunctions, equality in Courant's bound is impossible in dimensions $d\geq 2$.

The proof involves two principal ingredients:

• Faber–Krahn Inequality: Provides a uniform lower bound for the principal Dirichlet eigenvalue in terms of the domain’s area (or volume), with equality only for balls.
• Weyl’s Law: Describes the asymptotic density of eigenvalues, $N(\lambda)\sim C_d |\Omega| \lambda^{d/2}$ as $\lambda\to\infty$, where $C_d = (2\pi)^{-d}\omega_d$.

By applying the Faber–Krahn inequality to each nodal domain, normalizing areas, and summing over all nodal domains, combined with the asymptotic relation between eigenvalue labeling and the spectrum via Weyl’s law, Pleijel’s original argument gives a universal bound strictly below $1$ for $\limsup_{n\to\infty}\mu(u_n)/n$.

2. Quantitative Refinements and Improved Bounds

Recent work revisited the sharpness of the upper bound, as the equality in the Faber–Krahn inequality only holds for disk-like domains. A quantitative enhancement, involving a stability version of the Faber–Krahn inequality, allows one to exploit the geometric “penalty” for departure from the optimal (disk-like) shape (Bourgain, 2013). Specifically,

$$\lambda_1(G) \geq \lambda_1(G_0)\left[ 1 + 250\min\Bigl(1 - \frac{r_i(G)}{r_0(G)}, 2(1 - (\frac{r_i(G)}{r_0(G)})^2)\Bigr)\right],$$

where $G_0$ is the disk of the same area as $G$, and $r_i(G), r_0(G)$ are the inradius and radius of $G_0$, respectively.

Furthermore, the argument incorporates an analysis of the packing density of nearly optimal (disk-like) nodal domains—the densest possible congruent disk packing in $\mathbb{R}^2$ has density $\pi/\sqrt{12}\approx 0.9069$—to show that geometric constraints preclude a “full” covering of the domain by almost-minimizing disks. Combining refined stability estimates and packing constraints yields an explicit (albeit minuscule) improvement in the bound,

$$\limsup_{n\to\infty}\frac{\mu(u_n)}{n}< \frac{4}{j^2} - 3\cdot 10^{-9}.$$

Numerically small, the improvement is conceptually significant because it rigorously rules out the original constant’s optimality by quantifying the cumulative effect of deviations from optimal shapes (Bourgain, 2013).

3. Extensions to Other Geometries and Operators

3.1 Neumann, Robin, and Schrödinger Operators

The adaptation to Neumann and Robin boundary problems was addressed in (Léna, 2016, Hassannezhad et al., 2023). For Neumann Laplacians on domains with $C^{1,1}$-boundary, the reflection method and partition–of–unity arguments allow for the transfer of the Faber–Krahn argument to “bulk” and “boundary” nodal domains, yielding the same upper bound constant ($\gamma(d)$) as in the Dirichlet case, confirming a conjecture of Pleijel. In the Robin case, M. van den Berg and coauthors show that the improved bound holds even if the Robin parameter is allowed to be negative (Hassannezhad et al., 2023).

For Schrödinger operators with confining and (certain) decaying radial potentials, versions of the theorem hold, with the nodal count controlled by geometric–spectral constants depending both on the symmetry and the potential’s growth or decay (Charron, 2015, Charron et al., 2016, Charron et al., 2021).

3.2 Non-Euclidean and Sub-Riemannian Spaces

Generalizations extend to non-smooth settings such as RCD$(K,N)$ spaces, which encompass smooth manifolds, Alexandrov spaces, and metric measure spaces with lower Ricci curvature bounds (Ponti et al., 2023). The pleijel theorem holds in such settings by virtue of almost-Euclidean isoperimetric inequalities and a suitable Weyl law.

For sub-Riemannian Laplacians, the reduction to nilpotent group models (e.g., the Heisenberg group and its variants) allows one to define explicit local Weyl and Faber–Krahn constants (Frank et al., 21 Feb 2024, Qiu, 22 Oct 2025). The asymptotic nodal count is then governed by the ratio of these constants. Monotonicity and explicit calculation yield unconditional bounds $\gamma(G)<1$ for all but four exceptional $H$-type groups (Qiu, 22 Oct 2025); in exceptional cases, the asymptotic behavior is conjecturally determined by geometric input, such as the sharp isoperimetric constant (notably the Pansu conjecture).

4. Precise Constants and Extremal Domains

The sharpness and calculation of Pleijel constants for specific domains have been the subject of recent research (Bobkov, 2018). For the unit disk,

$\operatorname{Pl}(\mathrm{Disk}) = 0.4613019...,$

with the value obtained via a detailed analysis of the zeros of Bessel functions and their nodal counts. For rectangles, simple product structures yield

$$\operatorname{Pl}(\mathcal{R}_{a_1,\dots,a_N}) = \frac{2^N \Gamma(N/2+1)}{\pi^{N/2} N^{N/2}},$$

which is strictly decreasing in $N$. For rings and sectors, the constant is characterized in terms of the asymptotics of zeros of cross-products of Bessel functions.

The minimal spectral partition problem, closely linked to large-$k$ asymptotics, suggests that the optimal "tile" for high-$k$ partitions is often not the disk but the regular hexagon. The so-called “hexagonal conjecture” posits that in planar domains,

$$\limsup_{n\to\infty}\frac{\mu(u_n)}{n} \leq \frac{4\pi}{\lambda_1(\text{Hexa}_1)} \approx 0.677,$$

where $\lambda_1(\text{Hexa}_1)$ is the first Dirichlet eigenvalue of the unit-area regular hexagon (Helffer et al., 2015). For nodal-like (bipartite) partitions, even sharper bounds (e.g., $2/\pi\approx 0.6366$) are expected (Polterovich’s conjecture).

5. Courant-Sharp Eigenvalues and Complete Classifications

In all settings where precise analysis is possible—2D tori, rectangles, cubes, and symmetric domains—only finitely many Courant-sharp eigenvalues exist, and many works determine exactly which eigenvalues attain equality (Helffer et al., 2014, Léna, 2015, Bérard et al., 2015, Helffer et al., 2015, Helffer et al., 2015, Léna, 2015). For instance, on the square torus or the 3D cube, only a handful of low-index eigenvalues are Courant-sharp.

The method typically combines spectral asymptotics (Weyl law), the Faber–Krahn inequality, and exact nodal counting via symmetry and explicit eigenfunction formulas. In higher dimensions or on more complicated (sub-Riemannian or non-Euclidean) structures, finiteness and, in many cases, explicit bounds on the number of Courant-sharp eigenfunctions can be deduced by analytic and geometric techniques (Helffer et al., 2015, Léna, 2015, Frank et al., 21 Feb 2024, Qiu, 22 Oct 2025).

6. Connections to Integral Geometry and Isoperimetric Defects

Integral identities in convex geometry, such as Pleijel’s identity and its multidimensional versions, relate global quantities (e.g., perimeter, area) to integrals over geometric configurations (e.g., lines, chords) and play a role in analyzing isoperimetric defects and stability (Moseeva, 2022). These links reflect the underlying geometric structure of nodal partitions and manifest in the isoperimetric foundations of the Faber–Krahn inequality, central in all Pleijel-type arguments.

7. Generalizations and Future Directions

The roadmap emerging from recent works (Frank et al., 21 Feb 2024, Qiu, 22 Oct 2025) indicates that Pleijel’s theorem operates at the intersection of sharp spectral asymptotics (Weyl constants), geometric inequalities (Faber–Krahn and isoperimetric), and algebraic structures (e.g., nilpotent and H-type groups). Open problems include removing exceptional cases in the sub-Riemannian setting (dependent on sharp isoperimetric inequalities), understanding universality of the Pleijel constant for different boundary conditions (Hassannezhad et al., 2023), and mapping the implications for quantum chaos, minimal spectral partitions, and geometry-driven spectral optimization.

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Summary Table: Pleijel Constants and Results in Key Settings

Domain/Operator	LimSup Nodal Ratio	Number/Character of Courant-sharp eigenvalues
Planar Dirichlet Laplacian	$4/j^2 \approx 0.691$ (improved: $<4/j^2$)	Finitely many, low-index only
Planar Disk	$0.4613...$	Finitely many
Two-dimensional rectangle	$2/\pi$	Finitely many, computed explicitly
Flat 2D torus	Only $k=1,2,3,4,5$	Precise list
Cube (3D)	$9/(2\pi^2)$	Only first 2 eigenvalues
Heisenberg/H-type groups	$\gamma(G)<1$ (non-exceptional)	All but four explicit cases (exceptions investigated)
RCD spaces/uniform domains	$\gamma(N)<1$ (explicit)	Finitely many
Robin Neumann/Schrödinger operators	As for Dirichlet (modulo geometric or operator data)	Finitely many

These results provide a unified framework for understanding nodal domains’ asymptotics, tightly coupling spectral and geometric properties, across a wide swath of mathematical and physical settings.