About Bounds for Eigenvalues of the Laplacian with Density

Abstract

Let 𝑀 denote a compact, connected Riemannian manifold of dimension 𝑛 ∈ ℕ. We assume that 𝑀 has a smooth and connected boundary. Denote by 𝑔 and d𝑣𝑔, respectively, the Riemannian metric on 𝑀 and the associated volume element. Let Δ be the Laplace operator on 𝑀 equipped with the weighted volume form d𝑚:= e⁻ʰd𝑣𝑔. We are interested in the operator Lₕ⋅ :=e⁻ʰ⁽ᵅ⁻¹⁾(Δ⋅+α𝑔(∇h, ∇⋅)), where α > 1 and 𝘩 ∈ 𝐶²(𝑀) are given. The main result in this paper states the existence of upper bounds for the eigenvalues of the weighted Laplacian Lₕ with the Neumann boundary condition if the boundary is non-empty.