Pólya's conjecture for Dirichlet eigenvalues and its Ku–Ku–Tang generalization to the poly-Laplacian: pointwise Weyl-constant lower bounds on arbitrary bounded domains

The Pólya conjecture (1954) asserts that the $k$-th Dirichlet eigenvalue of the Laplacian on any bounded domain $\Omega\subset\mathbb{R}^n$ satisfies $\lambda_k\ge 4\pi^2(\omega_nV(\Omega))^{-2/n}k^{2/n}$, i.e. that the leading term of Weyl's asymptotic law is already a pointwise lower bound for every $k$; the generalized Pólya conjecture of Ku–Ku–Tang (1992) predicts the analogous bound $\lambda_k^{(l)}\ge(4\pi^2)^l(\omega_nV(\Omega))^{-2l/n}k^{2l/n}$ for the Dirichlet eigenvalues of the poly-Laplacian $(-\Delta)^l$ of every order $l\ge1$. The motivating source (Ji–Xu, arXiv:2508.04069) proves only improved lower bounds for sums of the first $k$ eigenvalues and explicitly leaves the pointwise inequalities open. Both conjectures remain open for arbitrary bounded domains: the pointwise bound is known only for tiling domains, balls, planar sectors, annuli, and a few related settings, while for general domains only the sharp averaged (Li–Yau type) inequalities are available. The task is to prove or disprove the pointwise Weyl-constant inequality for Dirichlet eigenvalues of the poly-Laplacian on arbitrary bounded domains, for every order $l\ge1$ and every index $k$.

Background

Spectral geometry studies how the eigenvalues of elliptic operators on a domain encode its geometry. For a bounded domain $\Omega\subset\mathbb{R}^n$, write $V(\Omega)$ for its $n$-dimensional volume and $\omega_n$ for the volume of the unit ball in $\mathbb{R}^n$. The Dirichlet eigenvalue problem for the Laplacian, $-\Delta u=\lambda u$ in $\Omega$ with $u=0$ on $\partial\Omega$, has a discrete spectrum $0<\lambda_1\le\lambda_2\le\cdots\to\infty$ (eigenvalues repeated with multiplicity); $\lambda_k$ is the square of the $k$-th resonant frequency of an idealized membrane fixed along the boundary. Weyl's law gives the asymptotic behavior $\lambda_k\sim 4\pi^2(\omega_nV(\Omega))^{-2/n}k^{2/n}$ as $k\to\infty$. In 1954 Pólya conjectured that this leading term is already a universal pointwise lower bound, $\lambda_k\ge 4\pi^2(\omega_nV(\Omega))^{-2/n}k^{2/n}$ for every $k$; he proved it in 1961 for planar tiling domains, i.e. planar domains whose congruent copies fill $\mathbb{R}^2$ without overlaps or gaps. The best general result is the Li–Yau inequality (1983), a sharp bound on averages, $\frac{1}{k}\sum_{i=1}^k\lambda_i\ge\frac{n}{n+2}4\pi^2(\omega_nV(\Omega))^{-2/n}k^{2/n}$, which falls short of the pointwise claim by the factor $\frac{n}{n+2}$; Melas (2003) strengthened it by a second term involving the moment of inertia of $\Omega$, and a long line of work has refined these averaged bounds. A recent breakthrough by Filonov–Levitin–Polterovich–Sher verified the conjecture for the disk (the first non-tiling planar domain), for arbitrary planar sectors, and, in the Dirichlet case, for balls of every dimension (Inventiones mathematicae, 2023), using Bessel-phase estimates, lattice-point counting, and rigorous computer-assisted interval arithmetic (arithmetic on intervals with endpoints that carry guaranteed error bounds, so that numerical results become certificates); the same authors settled Dirichlet annuli (2026) and Neumann balls (2026), and Frank–Loss–Weidl (2009) proved the conjecture in the presence of a constant magnetic field. For a general bounded domain the pointwise conjecture has remained open since 1954.

For an integer $l\ge2$, the poly-Laplacian $(-\Delta)^l$ is the $l$-fold iterate of the Laplacian; the case $l=2$ is the biharmonic operator governing the vibration of a stiff plate. Its Dirichlet (or clamped) eigenvalue problem imposes $(-\Delta)^lu=\lambda u$ in $\Omega$ with $u=\partial u/\partial\nu=\cdots=\partial^{l-1}u/\partial\nu^{l-1}=0$ on $\partial\Omega$, where $\nu$ is the outward unit normal; the spectrum $0<\lambda_1^{(l)}\le\lambda_2^{(l)}\le\cdots$ is again discrete, and Weyl's law gives $\lambda_k^{(l)}\sim(4\pi^2)^l(\omega_nV(\Omega))^{-2l/n}k^{2l/n}$. Ku–Ku–Tang (1992) proposed the generalized Pólya conjecture: the pointwise bound $\lambda_k^{(l)}\ge(4\pi^2)^l(\omega_nV(\Omega))^{-2l/n}k^{2l/n}$ for every $k$ (in their original form, for polynomial operators $\mathfrak{L}=\sum_ma_m(-\Delta)^m$ with $a_m\ge0$, the predicted bound is $\lambda_k\ge\sum_ma_m(4\pi^2/\omega_n^{2/n})^m(k/V(\Omega))^{2m/n}$, of which the pure poly-Laplacian is the principal case). What is known for $l\ge2$ is again only bounds on averages of eigenvalues: Cheng–Qi–Wei (2013) proved Li–Yau-type multi-term lower bounds for $\frac{1}{k}\sum_{i=1}^k\lambda_i^{(l)}$, Xu–Xu (2020) extended such inequalities to Cartan–Hadamard manifolds (complete, simply connected Riemannian manifolds of nonpositive sectional curvature), and Ji–Xu (arXiv:2508.04069, 2025) sharpens the correction coefficients of these sum bounds in all dimensions — its Theorem 1.1 for $n\ge5$ requires an explicit lower bound on $k$, and its unrestricted Theorem 1.2 requires $2l+n\ge6$. None of these works addresses the pointwise inequality. Thus, for both the classical Laplacian ($l=1$) and the poly-Laplacian ($l\ge2$), whether the exact Weyl constant yields a pointwise lower bound on an arbitrary bounded domain is open.

Problem Statement

Let $\Omega\subset\mathbb{R}^n$ be an arbitrary bounded domain, let $V(\Omega)$ be its volume and $\omega_n$ the volume of the unit ball in $\mathbb{R}^n$, and let $0<\lambda_1^{(l)}\le\lambda_2^{(l)}\le\cdots$ be the Dirichlet eigenvalues of the poly-Laplacian $(-\Delta)^l$ with clamped boundary conditions $u=\partial u/\partial\nu=\cdots=\partial^{l-1}u/\partial\nu^{l-1}=0$ on $\partial\Omega$ (for arbitrary bounded domains these conditions are understood in the standard way via the Friedrichs realization, i.e. the closure $H^l_0(\Omega)$ of $C_c^\infty(\Omega)$, which has discrete spectrum and satisfies the Weyl asymptotics stated above), listed with multiplicity. Prove or disprove: for every integer $l\ge1$, every integer $n\ge1$, every bounded domain $\Omega$, and every $k\ge1$,

$$\lambda_k^{(l)}(\Omega)\ \ge\ (4\pi^2)^{l}\,\bigl(\omega_nV(\Omega)\bigr)^{-2l/n}\,k^{2l/n}.$$

For $l=1$ this is the classical Pólya conjecture of 1954 (in counting-function form: the Dirichlet eigenvalue counting function of any bounded domain is bounded above by the leading Weyl term at every energy). For $l\ge2$ it is the generalized Pólya conjecture of Ku–Ku–Tang (1992), of which the corresponding statement for polynomial operators $\mathfrak{L}=\sum_ma_m(-\Delta)^m$ with $a_m\ge0$ — namely $\lambda_k(\mathfrak{L})\ge\sum_ma_m(4\pi^2/\omega_n^{2/n})^m(k/V(\Omega))^{2m/n}$ — is the original formulation, the pure poly-Laplacian being its principal case. A complete answer settles the conjecture either for all orders $l\ge1$ (proof) or by exhibiting one bounded domain, one order $l\ge1$, and one index $k$ for which the displayed inequality fails (refutation).

References

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