Pólya’s conjecture states that all eigenvalues of the Dirichlet Laplacian on bounded Euclidean domains are above the first term in the corresponding Weyl's law. The papers below are about the conjecture for low eigenvalues [F19], non-tiling domains satisfying the conjecture and related inequalities on manifolds [FS23,FMS25], relation to extremal domains [FLP21], and extensions to the case of Robin boundary conditions [AFK13,FK21].
[AFK13] Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian. ESAIM: Control, Optimisation and Calculus of Variations 19 (2013), 438–459 (with Pedro R. S. Antunes and James B. Kennedy).
[FMS25] Pólya-type inequalities on spheres and hemispheres, Ann. Inst. Fourier (Grenoble) 75 (2025), 979–1051 (with Jing Mao and Isabel Salavessa).
[FS23] Families of non-tiling domains satisfying Pólya’s conjecture, J. Math. Phys. 64 (2023), 121503, 7pp (with Isabel Salavessa).
[FK21] Extremal domains and Pólya-type inequalities for the Robin Laplacian on rectangles and unions of rectangles, Int. Math. Res. Not. IMRN (2021), no. 18, 13730–13782 (with James B. Kennedy).
[FLP21] Optimal unions of scaled copies of domains and Pólya's conjecture. Ark. Math. 59 (2021), 11–51 (with Jean Lagacé and Jordan Payette).
[F19] A remark on Pólya's conjecture at low frequencies. Arch. Math. (Basel) 112 (2019), 305–311.