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Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary

Lashi Bandara, Medet Nursultanov, Julie Rowlett

TL;DR

This work proves Weyl-type eigenvalue asymptotics for Laplacians on compact manifolds with smooth boundary equipped with rough (bounded, measurable) metrics, extending to weighted problems with sign-changing weights ρ∈L^β (β>n/2). The authors develop a wholly functional-analytic approach inspired by Birman and Solomjak, circumventing heat-kernel methods which are unavailable due to the lack of a canonical distance, by localizing to Euclidean patches and reassembling the spectral data. They show that the weighted Laplacian with admissible boundary conditions on a rough manifold has a discrete spectrum consisting of positive and negative eigenvalues {−λ_j^−, λ_j^+}, and they establish the global Weyl law lim_{k→∞} λ_k^{±} k^{2/n} = ((ω_n)/(2π)^n)^{2/n} (∫_{M^{±}} |ρ|^{n/2} dμ_g)^{2/n}, where M^{±} = {x∈M: ±ρ(x)>0}. The work builds on the Birman–Solomjak theory for non-smooth coefficients and highlights rough metrics as geometric invariances of the Kato square root problem, with potential applications to broader spectral problems on singular geometries.

Abstract

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.

Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary

TL;DR

This work proves Weyl-type eigenvalue asymptotics for Laplacians on compact manifolds with smooth boundary equipped with rough (bounded, measurable) metrics, extending to weighted problems with sign-changing weights ρ∈L^β (β>n/2). The authors develop a wholly functional-analytic approach inspired by Birman and Solomjak, circumventing heat-kernel methods which are unavailable due to the lack of a canonical distance, by localizing to Euclidean patches and reassembling the spectral data. They show that the weighted Laplacian with admissible boundary conditions on a rough manifold has a discrete spectrum consisting of positive and negative eigenvalues {−λ_j^−, λ_j^+}, and they establish the global Weyl law lim_{k→∞} λ_k^{±} k^{2/n} = ((ω_n)/(2π)^n)^{2/n} (∫_{M^{±}} |ρ|^{n/2} dμ_g)^{2/n}, where M^{±} = {x∈M: ±ρ(x)>0}. The work builds on the Birman–Solomjak theory for non-smooth coefficients and highlights rough metrics as geometric invariances of the Kato square root problem, with potential applications to broader spectral problems on singular geometries.

Abstract

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.

Paper Structure

This paper contains 17 sections, 18 theorems, 132 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Weyl's law in singular geometric settings
  3. Strategy and structure of the paper
  4. The rough Riemannian manifold setting
  5. Rough metrics in harmonic analysis
  6. Geometric examples
  7. Analytic preliminaries
  8. Notation
  9. Dirichlet Forms and operators
  10. Laplacian associated to admissible boundary conditions
  11. Examples of boundary conditions
  12. Proof of the main results
  13. The statement of the problem
  14. Eigenvalue asymptotics for the weighted Laplace equation on Euclidean domains
  15. An auxiliary problem
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $M$ be a smooth compact manifold of dimension $\geq 2$ with smooth boundary, and let $g$ be a rough metric on $M$. Then, the eigenvalues of evprob are discrete with finite dimensional eigenspaces with positive and negative eigenvalues, $\{ - \lambda_j ^- (\mathcal{W}); \lambda_j ^+ (\mathcal{W}) Moreover, they satisfy the Weyl asymptotic formula Above, $M^\pm := \{ x \in M : \pm \rho(x) > 0 \

Figures (2)

  • Figure 1: The witch's hat sphere metric.
  • Figure 2: The eigenvalues and functions, together with their respective geometric domains and Dirichlet forms are organised in the above table.

Theorems & Definitions (42)

  • Theorem 1.1: Weyl asymptotics for weighted Laplace equation with admissible boundary conditions
  • Corollary 1.2: Classical Weyl asymptotics
  • Definition 2.1: Rough metric
  • Remark 2.2
  • Example 2.3: Rough metrics arising from Lipschitz graphs
  • Example 2.4: Manifolds with geometric cones
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Definition 3.2: Admissible boundary condition
  • ...and 32 more