Two New Weyl-type Bounds for the Dirichlet Laplacian
نویسنده
چکیده
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 λ ≥ λ1, one has N(λ) > 2 n+ 2 1 Hn (λ− λ1) n/2 λ −n/2 1 , and N(λ) > „ n+ 2 n+ 4 «n/2 1 Hn (λ− (1 + 4/n) λ1) n/2 λ −n/2 1 , where Hn = 2 n j2 n/2−1,1 J2 n/2 (jn/2−1,1) is a constant which depends on n, the dimension of the underlying space, and Bessel functions and their zeros.
منابع مشابه
. SP ] 2 4 M ay 2 00 7 Differential inequalities for Riesz means and Weyl - type bounds for eigenvalues 1
We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z) := ∑ k (z − λk) σ +. Here {λk} ∞ k=1 are the ordered eigenvalues of the Laplacian on a bounded domain Ω ⊂ Rd, and x+ := max(0, x) denotes the positive part of the quantity x. As corollaries of these inequalities, we derive Weyl-type bounds on λk, on averages such as λ...
متن کاملDifferential inequalities for Riesz means and Weyl-type bounds for eigenvalues
We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z) := ∑ k (z − λk) σ +. Here {λk} ∞ k=1 are the ordered eigenvalues of the Laplacian on a bounded domain Ω ⊂ Rd, and x+ := max(0, x) denotes the positive part of the quantity x. As corollaries of these inequalities, we derive Weyl-type bounds on λk, on averages such as λ...
متن کاملUniversal bounds and semiclassical estimates for eigenvalues
We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues {λk} that are analogous to those known for Schrödinger operators and the Dirichlet Laplacian, on which the operators of interest are modeled. In addition we produce inequalities that are new even in the model case. These...
متن کاملOn Riesz Means of Eigenvalues
In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace, Legendre, Weyl, and Mellin, and the Riemann-Liouville fractional transform. We also prove new universal eigenvalue inequalities and monotonicity principles for Diric...
متن کاملPositive solution for Dirichlet $p(t)$-Laplacian BVPs
In this paper we provide existence results for positive solution to Dirichlet p(t)-Laplacian boundary value problems. The sublinear and superlinear cases are considerd.
متن کامل