Spectrum of the Laplacian with weights

Abstract

Given a compact Riemannian manifold (M, g) and two positive functions ρ and σ, we are interested in the eigenvalues of the Dirichlet energy functional weighted by σ, with respect to the L 2 inner product weighted by ρ. Under some regularity conditions on ρ and σ, these eigenvalues are those of the operator −ρ −1 div(σ∇u) with Neumann conditions on the boundary if ∂M = ∅. We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is preserved.2010 Mathematics Subject Classification. 35P15, 58J50.

Eigenvalues of elliptic operators with density

“…Proof of Theorem 3.4. The proof is based on the general method described by Grigor'yan, Netrusov and Yau in [27] (see Theorem 3.1; see also [14,15] for the case of the Laplace operator). In particular, we will build a suitable family of disjointly supported test functions with controlled Rayleigh quotient.…”

“…In view of the physical interpretation of problem (1.1) when m = 1 and N = 1 or N = 2, it is very natural to ask whether it is possible to redistribute a fixed amount of mass on a string (of fixed length) or on a membrane (of fixed shape) such that all the eigenvalues become arbitrarily large when the body is left free to move, or, on the contrary, if there exists uniform upper bounds for all the eigenvalues. As highlithed in [14], uniform upper bounds with mass constraint exist if N ≥ 2. In this paper, by using the techniques of [14] we prove that if N ≥ 2m, uniform upper bounds exist (see Theorem 3.4), namely we prove that if N ≥ 2m…”

“…As highlithed in [14], uniform upper bounds with mass constraint exist if N ≥ 2. In this paper, by using the techniques of [14] we prove that if N ≥ 2m, uniform upper bounds exist (see Theorem 3.4), namely we prove that if N ≥ 2m…”

Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions

“…the last equality following from an integration by parts in (17). This contradiction yields that v ij ≡ 0 in E. Hence x i ∂ x j ϕ ≡ x j ∂ x i ϕ for all i = j, which implies that ϕ is radially symmetric inside E. In other words, there exists a function U such that ϕ(x) = U (|x|), for all x ∈ E.…”

About Bounds for Eigenvalues of the Laplacian with Density

“…A main interest is to investigate the interplay between the geometry of (M, g) and the effect of the weights, looking at the behaviour of λ k (ρ, ρ α ), among densities ρ of fixed total mass. The more general problem where the Dirichlet energy functional is weighted by a positive function σ, not necessarily related to ρ is presented by Colbois and El Soufi in [4].…”