Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues

Abstract

For smooth bounded domains in R n , we prove upper and lower L 2 bounds on the boundary data of Neumann eigenfunctions, and prove quasiorthogonality of this boundary data in a spectral window. The bounds are tight in the sense that both are independent of eigenvalue; this is achieved by working with an appropriate norm for boundary functions, which includes a 'spectral weight', that is, a function of the boundary Laplacian. This spectral weight is chosen to cancel concentration at the boundary that can happen for 'whispering gallery' type eigenfunctions. These bounds are closely related to wave equation estimates due to Tataru.Using this, we bound the distance from an arbitrary Helmholtz parameter E > 0 to the nearest Neumann eigenvalue, in terms of boundary normalderivative data of a trial function u solving the Helmholtz equation (∆−E)u = 0. This 'inclusion bound' improves over previously known bounds by a factor of E 5/6 . It is analogous to a recently improved inclusion bound in the Dirichlet case, due to the first two authors.Finally, we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of Neumann eigenpairs on smooth planar domains. We show that the new inclusion bound improves the relative accuracy in a computed Neumann eigenvalue (around the 42000th) from 9 digits to 14 digits, with little extra effort.

On the decay rate for the wave equation with viscoelastic boundary damping

“…For example if the boundary of ∂Ω is of class C ∞ then both estimates hold with α = 2/3. See [3] for this result.…”

“…Let us suppose that Ω has a C ∞ boundary and let k = k β,ε for some ε > 0 and 0 < β < 1. By [3] we know that (23) is satisfied for α = 2/3. Thus by (26) and Theorem 9 we have…”

“…and (τ → 1 1+τ ) ∈ L 1 ν ∩ L 2 ν (0, ∞) by (2). 3 The quadratic forms E and E 1 turn H and H 1 into Hilbert spaces respectively.…”

TheL2behavior of eigenfunctions near the glancing set

“…In particular, for respectively Dirichlet and Neumann eigenfunctions we have the sharp estimates h∂ ν u| ∂M L 2 (∂M ) ≤ C, u| ∂M L 2 (∂M ) ≤ Ch −1/3 . Moreover, in [BHT15] the authors show that for Neumann eigenfunctions (4) (1 + h 2 ∆ ∂M ) 1/2 + u| ∂M L 2 (∂M ) ≤ C. The authors also show that the power 1/2 in (4) is optimal in the sense that there are Neumann eigenfunctions such that replacing 1/2 by ρ < 1/2 may result in an L 2 norm that is not uniformly bounded.…”

Localization and landscape functions on quantum graphs

“…[32, §X.9], [14, §2], §2. See also [13,5,34] for other approaches to pointwise bounds on eigenfunctions. In [15] Davies showed that hypercontractive estimates can be adapted to the case of quantum graphs, as we shall recall in Proposition 2.1, below.…”