Two new Weyl-type bounds for the Dirichlet Laplacian

الأصل · EN

In this paper, we prove two new Weyl-type upper estimates for the eigenvalues of the Dirichlet Laplacian. As a consequence, we obtain the following lower bounds for its counting function. For ≥ ₁, one has N() > 2/n+2 1/Hₙ (-₁)ⁿ/² ₁⁻ⁿ/², and N() > (n+2/n+4)ⁿ/² 1/Hₙ (-(1+4/n) ₁)ⁿ/² ₁⁻ⁿ/², where Hₙ=2 njₙ/₂₋₁,₁² Jₙ/₂²(jₙ/₂₋₁,₁) is a constant which depends on n, the dimension of the underlying space, and Bessel functions and their zeros.