Universal Bounds for the Low Eigenvalues of Neumann Laplacians inNDimensions

The authors consider bounds on the Neumann eigenvalues of the Laplacian on domains in IRⁿ in the light of their recent results on Dirichlet eigenvalues, in particular, their proof of the Payne-Pólya–Weinberger conjecture via spherical rearrangement. They prove the bound 1 / μ ₁ + 1 / μ ₂ ≥ A / 2π for the first two nonzero Neumann eigenvalues for an arbitrary bounded domain Ω in two dimensions and also the stronger (and optimal) bound μ ₂ ≤ π (j'1,1 )^2 / A for domains having a 4-fold rotational symmetry. (Here (j'1,1 ) ≈ 1.84118 denotes the first positive zero of the derivative of the Bessel function J₁ (x) and A is the area of the domain Ω.) The authors also obtain analogues of these results for domains in IRⁿ. Previous results in this vein are due to Szegö, who proved μ ₁ ≤ π (j'1,1 )² / A and 1 / μ ₁ + 1 / μ ₂ 2A / π (j'1,1 )² for simply connected domains in IR², and to Weinberger, who proved the general resultμ ₁ ≤ (Cₙ / |Ω |)^2 / n p_n / 2,1² for arbitrary domains in IRⁿ (here Cₙ = π ^n / 2 Cₙ = π ^n / 2 Γ (n / 2 + 1)=$ volume of the unit ball in $IR^n $, and $pν ,k $ denotes the kth positive zero of the derivative of $x1 - ν J_ν (x)$, where $J_ν (x)$ represents the standard Bessel function of the first kind of order v).