Lower bounds on Steklov eigenfunctions for a class of curvilinear polygons
Abstract
Steklov eigenfunctions, {ϕσj}j=0∞, are the eigenfunctions of the Dirichlet to Neumann map on a bounded domain with eigenvalue σj. This paper considers Steklov eigenfunctions and eigenvalues on a family of curvilinear polygons, Ωα whose boundaries are smooth away from a single corner of angle α. We (i) prove optimal lower bounds on the decay of the harmonic extension, uσj, of ϕσj away from the boundary of Ωα and (ii) give optimal lower bounds on the degree to which the high energy Steklov spectrum of Ωα∖ H1 may differ from that of Ωα∖ H0 when H1 and H0 are isometric. This shows that the corresponding estimates in recent work by the authors together with Malagutti are optimal. The proof in this paper was obtained through an interaction with ChatGPT 6 Astra: While writing the work with Malagutti, the authors became curious about optimal error bounds for the difference between the Steklov spectra on a curvilinear polygon, P, and the spectrum of any operator which knows only local information about ∂ P. The authors suggested to ChatGPT to remove two isometric, smooth domains from a curvilinear polygon. ChatGPT suggested the ideas for a proof based on a regularized trace and the singularity in the kernel of the DtN map near a corner. The authors then suggested a simpler, conformal approach, the method for handling small angles, and the idea to obtain lower bounds on Steklov eigenfunctions. The paper is entirely written by the authors and the argument has been significantly modified, clarified, and simplified from the ChatGPT ideas.
Related
- Sharp Bounds for Nodal Sets of Steklov Eigenfunctions on Smooth Surfaces
- Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface
- Tilings of Closed Surfaces by Congruent Most Curvilinear Polygons
- A sharp curvature lower bound for the first exterior p-harmonic Steklov eigenvalue
- Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes
BibTeX
@misc{galkowski2026lower,
title = {{Lower bounds on Steklov eigenfunctions for a class of curvilinear polygons}},
author = {Jeffrey Galkowski and Ruoyu P. T. Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31387},
eprint = {2610.31387},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31387},
}
RIS
TY - UNPB
TI - Lower bounds on Steklov eigenfunctions for a class of curvilinear polygons
AU - Jeffrey Galkowski
AU - Ruoyu P. T. Wang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31387
UR - https://arxiv.org/abs/2610.31387
AB - Steklov eigenfunctions, {ϕ_σ_j}_j=0^∞, are the eigenfunctions of the Dirichlet to Neumann map on a bounded domain with eigenvalue σ_j. This paper considers Steklov eigenfunctions and eigenvalues on a family of curvilinear polygons, Ω_α whose boundaries are smooth away from a single corner of angle α. We (i) prove optimal lower bounds on the decay of the harmonic extension, u_σ_j, of ϕ_σ_j away from the boundary of Ω_α and (ii) give optimal lower bounds on the degree to which the high energy Steklov spectrum of Ω_α∖ H_1 may differ from that of Ω_α∖ H_0 when H_1 and H_0 are isometric. This shows that the corresponding estimates in recent work by the authors together with Malagutti are optimal. The proof in this paper was obtained through an interaction with ChatGPT 6 Astra: While writing the work with Malagutti, the authors became curious about optimal error bounds for the difference between the Steklov spectra on a curvilinear polygon, P, and the spectrum of any operator which knows only local information about ∂ P. The authors suggested to ChatGPT to remove two isometric, smooth domains from a curvilinear polygon. ChatGPT suggested the ideas for a proof based on a regularized trace and the singularity in the kernel of the DtN map near a corner. The authors then suggested a simpler, conformal approach, the method for handling small angles, and the idea to obtain lower bounds on Steklov eigenfunctions. The paper is entirely written by the authors and the argument has been significantly modified, clarified, and simplified from the ChatGPT ideas.
ER -