On Pleijel-type nodal domain bounds for the $p$-Laplacian

Abstract: We provide an upper estimate à la Pleijel on the asymptotic number of nodal domains for eigenfunctions corresponding to the cogenus eigenvalues ${λ<em>k(p;Ω)}$ of the $p$-Laplacian in a bounded domain $Ω$, and identify regimes when the number of nodal domains of the $k$-th eigenfunction is less than $k$ as $k \to +\infty$. As auxiliary results, which also have independent interest, we provide a useful characterization of the cogenus eigenvalues implying their continuity with respect to $p$, justify the Weyl law, and prove the inequality $$λ_2(p;B) \leq \dots \leq λ</em>{N+1}(p;B) \leq λ<em>\ominus(p)$$ in an $N$-dimensional ball $B$, where $λ</em>\ominus(p)$ is an eigenvalue whose eigenfunction has a central section of $B$ as its nodal set.

• The paper provides sharp asymptotic Pleijel-type nodal domain bounds for the p-Laplacian using cogenus and variational techniques.
• It applies Weyl law and geometric inequalities to derive explicit numerical bounds in dimensions 2–4 with optimal lattice packings.
• The study demonstrates sublinear growth of nodal domains and explores continuity through regularization of cogenus eigenvalue indices.

Pleijel-Type Nodal Domain Bounds for the $p$-Laplacian: An Expert Overview

Introduction and Context

The paper "On Pleijel-type nodal domain bounds for the $p$-Laplacian" (2606.31305) addresses the asymptotic distribution and behavior of nodal domains associated with eigenfunctions of the $p$-Laplacian on bounded domains, with a rigorous focus on eigenvalues indexed by the cogenus. This research generalizes classical results known for linear operators—specifically the Laplacian ($p = 2$)—to the nonlinear, quasilinear setting ($p \neq 2$), providing new tools and sharp bounds for understanding eigenfunction structure at high index.

Spectral Framework: $p$-Laplacian and Cogenus Eigenvalues

The core object of study is the Dirichlet $p$-Laplacian eigenproblem on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^N$: $$-\Delta_p u = \lambda |u|^{p-2}u \quad\text{in } \Omega, \quad u = 0 \text{ on } \partial\Omega,$$ where $\Delta_p u = \text{div}(|\nabla u|^{p-2} \nabla u)$ and $p$0. The focus is on variationally defined (minimax) eigenvalues, with a particular interest in those indexed using the cogenus, $p$1, constructed via continuous odd maps from higher-dimensional spheres into the relevant Sobolev sphere. The paper establishes equivalence between the cogenus and alternative minimax characterizations, verifies the continuity of cogenus eigenvalues with respect to $p$2, and justifies their suitability for nodal domain analysis.

Nodal Domain Estimates and Pleijel-Type Bounds

Central to the analysis is the investigation of the number of nodal domains, $p$3, for the $p$4th cogenus eigenfunction $p$5. While the classical Courant nodal domain theorem confirms the bound $p$6 in the linear case, and Drábek–Robinson [DR] showed that for $p$7, $p$8, this paper seeks substantially sharper bounds in the asymptotic regime as $p$9, specifically: $p$0

Building analogously to Pleijel's approach, the paper develops a general upper bound scheme for $p$1 based on variational characterizations, geometric inequalities (notably the Faber–Krahn inequality), and Weyl-type asymptotics for the spectral sequence. It exposes a delicate dependence of the nodal domain count on the Weyl constant $p$2 associated with the $p$3-Laplacian.

Key result: For all $p$4 and bounded Lipschitz domains $p$5,

$p$6

where $p$7 is the unit ball, and $p$8 its principal Dirichlet eigenvalue. Explicit sharp upper bounds are provided in dimensions 2–4 using optimal lattice packings.

Asymptotic Techniques and Sharper Nodal Bounds

Weyl Law and $p$9-Dependence

The proof of improved asymptotic nodal domain bounds heavily leverages the Weyl law for the $p = 2$0-Laplacian, recently established for cogenus-type eigenvalues. The law asserts, for large $p = 2$1,

$p = 2$2

for an explicit constant $p = 2$3. The analysis employs quantitative estimates of $p = 2$4 derived from lattice tilings and sphere-packing considerations.

Explicit Numerical Bounds

The consequences are explicit: in low dimensions and for certain lattices, using the best sphere packing densities $p = 2$5, the paper establishes

$p = 2$6

These bounds are uniform in $p = 2$7 for domains associated with optimal lattice packings and underline that—unlike the unconditional Courant bound—the ratio of nodal domains to index for large $p = 2$8 is strictly less than 2 and, depending on $p = 2$9, may be strictly less than 1.

Regimes of Strict Inequality

Notably, the research identifies two parameter regimes where the asymptotic nodal domain ratio satisfies: $p \neq 2$0 and also in the limit $p \neq 2$1.

For $p \neq 2$2 near 2, continuity arguments and spectral perturbation yield that the Pleijel's constant from the Laplacian governs the behavior to leading order, affirming the strict sub-unity of $p \neq 2$3 in an open neighborhood around $p \neq 2$4. As $p \neq 2$5, the Cheeger constants of balls and half-balls control the relevant eigenvalues, resulting in asymptotic ratios such as 0.91424 (in $p \neq 2$6) and 0.96969 (in $p \neq 2$7), both < 1.

Auxiliary Contributions

Regularized Cogenus and Continuity

A technical challenge handled in the paper is the regularization of the cogenus index—a crucial step since the classical cogenus does not satisfy topological regularity conditions required for perturbation and continuity results. The paper extends Coffman's approach to infinite dimensions, showing that the regularized cogenus provides an appropriate index for spectral and nodal analysis, and that all relevant cogenus-based eigenvalue characterizations are equivalent.

Multiplicity Results and Symmetry

A further auxiliary result gives a detailed analysis of the multiplicity of low-index $p \neq 2$8-Laplacian eigenvalues in symmetric domains, showing that for the $p \neq 2$9-dimensional ball, certain eigenvalues up to index $p$0 are bounded above by the smallest eigenvalue whose nodal set is a central section.

Comparison with Other Indices

The paper briefly addresses the status of similar results for eigenvalues indexed by the Krasnoselskii genus and cohomological index, noting that while Courant-type bounds generalize, the Weyl law for genus eigenvalues remains open due to unknown subadditivity properties.

Implications and Future Directions

This work substantially advances understanding of nodal domain structure for nonlinear elliptic operators. Its main theoretical implication is that, even in the absence of unique continuation and with nonlinear effects, asymptotics reminiscent of the classical Pleijel theorem persist and even improve in certain regimes. This sharpens expectations for the geometry of high-index eigenfunctions for quasilinear problems and supports the use of geometric and topological analysis in spectral problems beyond the linear setting.

Further, the work underscores a rich interplay between optimal packings, geometric inequalities (such as Faber–Krahn and isoperimetric bounds), and spectral properties, suggesting fruitful directions for extending these results to more general nonlinear operators, non-Euclidean spaces, and possibly to non-selfadjoint or indefinite problems.

The explicit connection of specific geometric constants (Weyl constants, Cheeger constants, packing densities) to spectral and nodal properties inspires potential investigations into their computation or estimation, both analytically and numerically, in higher dimensions and more complex geometries.

Conclusion

The paper establishes, for the first time with generality, strong Pleijel-type bounds on the nodal domain count for nonlinear $p$1-Laplacian eigenfunctions, proving that for high-index eigenfunctions, the number of nodal domains grows sublinearly in $p$2 in many regimes and obeys strict, dimension- and geometry-dependent bounds. The techniques, based on regularized cogenus indices, Weyl asymptotics, and geometric measure theory, enable robust extensions of classical spectral theory to highly nonlinear settings, laying foundational groundwork for further analytic and computational explorations in nonlinear PDE spectral geometry.