On the Isoperimetric Inequality for the Third Robin Eigenvalue on Surfaces
Abstract
We investigate the third Robin eigenvalue on simply connected domains of area A in surfaces with Gaussian curvature bounded above by K. For K>0, we discover a transition at the critical parameter α=−4π+KA: two equal geodesic disks give a sharp upper bound when α=−4π or −4π+KA≤α≤4π, but the two disk bound fails for −4π<α<−4π+KA. In the latter range, we establish a new upper bound involving a single geodesic disk with a modified Robin parameter and prove its sharpness in an explicit subrange. The sharp two-disk inequality persists for nonpositive curvature bounds, and Steklov--Helmholtz analogues are also obtained. Both the two-disk and one-disk bounds arise from a unified two-pole Green-function method, with the one-disk regime emerging from a degenerate area split that produces the 2π shift in the effective Robin parameter, while sharpness in the stated subrange is established by boundary oscillations that redistribute Robin boundary mass.
Related
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- A self-contained proof of the variational characterization of the principal eigenvalue of −Δ+q under general Robin boundary conditions of non-classical type
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BibTeX
@misc{wang2026isoperimetric,
title = {{On the Isoperimetric Inequality for the Third Robin Eigenvalue on Surfaces}},
author = {Xiaoliang Wang and Zuoqin Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34134},
eprint = {2610.34134},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34134},
}
RIS
TY - UNPB
TI - On the Isoperimetric Inequality for the Third Robin Eigenvalue on Surfaces
AU - Xiaoliang Wang
AU - Zuoqin Wang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34134
UR - https://arxiv.org/abs/2610.34134
AB - We investigate the third Robin eigenvalue on simply connected domains of area A in surfaces with Gaussian curvature bounded above by K. For K>0, we discover a transition at the critical parameter α=−4π+KA: two equal geodesic disks give a sharp upper bound when α=−4π or −4π+KA≤α≤4π, but the two disk bound fails for −4π<α<−4π+KA. In the latter range, we establish a new upper bound involving a single geodesic disk with a modified Robin parameter and prove its sharpness in an explicit subrange. The sharp two-disk inequality persists for nonpositive curvature bounds, and Steklov--Helmholtz analogues are also obtained. Both the two-disk and one-disk bounds arise from a unified two-pole Green-function method, with the one-disk regime emerging from a degenerate area split that produces the 2π shift in the effective Robin parameter, while sharpness in the stated subrange is established by boundary oscillations that redistribute Robin boundary mass.
ER -