Spectral analysis and shape optimization of Steklov--Dirichlet eigenvalues on shell-type domains
Abstract
We investigate the Steklov--Dirichlet eigenvalues on shell-type domains in ℝn, n≥ 2, where the Steklov and Dirichlet conditions are imposed on the outer and the inner boundaries, respectively. We introduce the Green function associated with the problem, which is well-defined for spectral parameters outside the spectrum and exhibits blow-up behavior at eigenvalues, and establish its spectral representation. We also examine its differentiability and derive its shape derivative under translations of the inner boundary. Using these results and a layer potential approach, we propose a computational framework for optimizing the eigenvalue with respect to the position of the inner boundary, incorporating an eigenvalue estimator for which a local error bound is established. Numerical experiments for various inner and outer boundary shapes demonstrate the effectiveness of the proposed approach in identifying configurations that maximize the first eigenvalue.
Related
- Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes
- Sharp upper bounds for Steklov eigenvalues on warped products
- Isodiametric-type upper bounds for Steklov eigenvalues
- Isoperimetric bounds for the weighted Neumann and Steklov eigenvalues in dimensions three through six
- Scattering for the Dirichlet-to-Neumann map and the Steklov spectrum
BibTeX
@misc{han2026spectral,
title = {{Spectral analysis and shape optimization of Steklov--Dirichlet eigenvalues on shell-type domains}},
author = {Dohyeong Han and Woojoo Lee and Mikyoung Lim},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04410},
eprint = {2611.04410},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04410},
}
RIS
TY - UNPB
TI - Spectral analysis and shape optimization of Steklov--Dirichlet eigenvalues on shell-type domains
AU - Dohyeong Han
AU - Woojoo Lee
AU - Mikyoung Lim
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04410
UR - https://arxiv.org/abs/2611.04410
AB - We investigate the Steklov--Dirichlet eigenvalues on shell-type domains in ℝ^n, n≥ 2, where the Steklov and Dirichlet conditions are imposed on the outer and the inner boundaries, respectively. We introduce the Green function associated with the problem, which is well-defined for spectral parameters outside the spectrum and exhibits blow-up behavior at eigenvalues, and establish its spectral representation. We also examine its differentiability and derive its shape derivative under translations of the inner boundary. Using these results and a layer potential approach, we propose a computational framework for optimizing the eigenvalue with respect to the position of the inner boundary, incorporating an eigenvalue estimator for which a local error bound is established. Numerical experiments for various inner and outer boundary shapes demonstrate the effectiveness of the proposed approach in identifying configurations that maximize the first eigenvalue.
ER -