Universal lower bounds for Dirichlet eigenvalues

Let Ω ⊂ Rᵈ be a bounded domain and let λ₁, λ₂, denote the sequence of eigenvalues of the Laplacian subject to Dirichlet boundary conditions. We consider inequalities for λₙ that are independent of the domain Ω. A well-known such inequality follows from the Berezin-Li-Yau approach λₙ ≥ d/d+2 4 π² n2/d (|Ω| ωd)2/d. The purpose of this paper is to point out a certain degree of flexibility in the Li-Yau approach. We use it to show that, for all 1 ≤ k ≤ n, λₙ ≥ d/d+2 4 π² n2/d (|Ω| ωd)2/d + k/n(λₙ - λₖ). This correction term is, at least asymptotically, not of lower order but comparable to the leading term. Moreover, the Berezin-Li-Yau inequality cannot be close to sharp on average. A family of universal inequalities follows as a byproduct: one example is, when $d=2$, the inequality 2 λₙ + λ₂ₙ ≥ 10 π n /|Ω|.