Removing Boundary Smoothness from the Hot-Spots Conclusion in the Simple-Eigenvalue Case
Abstract
We give a reduction from a smooth-domain theorem excluding interior critical points of first positive Neumann eigenfunctions to a non-strict hot-spots conclusion on arbitrary bounded simply connected planar domains with compact Neumann resolvent and simple first positive eigenvalue. The main analytic step is spectral convergence under exhaustion by conformal images of concentric disks. A reflection extension, conformal invariance of Dirichlet energy, and compactness of the Sobolev embedding give a direct proof. A truncation argument justifies square integrability of the transferred extensions without any boundary regularity. The smooth-domain theorem proved by OpenAI is an external input; the approximation argument is proved here in full. The paper was generated by ChatGPT with prompts from the listed author.
Related
- The Dirichlet problem with boundary data of fractional smoothness in domains with Ahlfors regular boundary
- A self-contained proof of the variational characterization of the principal eigenvalue of −Δ+q under general Robin boundary conditions of non-classical type
- Conformal Boundary Deformations under Ricci Lower Bounds: Eigenvalue Counterexamples
- Uniform boundary observability for periodic waves, I: up to the ε−1 eigenvalue scale
- Conformal boundary rigidity for simple Finsler metrics
BibTeX
@misc{burdzy2026removing,
title = {{Removing Boundary Smoothness from the Hot-Spots Conclusion in the Simple-Eigenvalue Case}},
author = {Krzysztof Burdzy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12622},
eprint = {2610.12622},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12622},
}
RIS
TY - UNPB
TI - Removing Boundary Smoothness from the Hot-Spots Conclusion in the Simple-Eigenvalue Case
AU - Krzysztof Burdzy
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12622
UR - https://arxiv.org/abs/2610.12622
AB - We give a reduction from a smooth-domain theorem excluding interior critical points of first positive Neumann eigenfunctions to a non-strict hot-spots conclusion on arbitrary bounded simply connected planar domains with compact Neumann resolvent and simple first positive eigenvalue. The main analytic step is spectral convergence under exhaustion by conformal images of concentric disks. A reflection extension, conformal invariance of Dirichlet energy, and compactness of the Sobolev embedding give a direct proof. A truncation argument justifies square integrability of the transferred extensions without any boundary regularity. The smooth-domain theorem proved by OpenAI is an external input; the approximation argument is proved here in full. The paper was generated by ChatGPT with prompts from the listed author.
ER -