Optimal location of small favourable regions for Robin eigenvalues with indefinite weights
Abstract
We consider the positive principal eigenvalue of an elliptic problem with a Robin boundary condition and bang--bang indefinite weight κ1D−1Ω∖ D, and ask where a favourable region D of prescribed small volume |D|=δ should be located. Put ε=δ1/N and τδ=αδε. We prove that there is a finite threshold τ*=τ*(N,κ), independent of the ambient domain, such that δ2/NΛδ(αδ)→Λℍ(τ) whenever τδ→τ<∞. If τ<τ*, optimal small regions concentrate at the boundary. If τ>τ*, their concentration centres move to distances much larger than ε from the boundary, and after recentring and rescaling the favourable sets converge in measure to the whole-space optimal ball. At τ=τ*, the half-space problem admits a compact boundary optimiser as well as minimising sequences escaping to infinity. In the finer regime τδ=τ*+σε+o(ε), we determine the first-order competition between the boundary and interior configurations. Tangential symmetry of compact threshold optimisers reduces the geometric correction to mean curvature. In particular, every fixed finite Robin coefficient is asymptotically in the boundary regime. Numerical experiments illustrate the boundary--interior transition, the three cases in the first-order selection law, and the curvature-dependent boundary location; for N=2 and κ=1 they suggest a transition near τ*≈3.2.
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BibTeX
@misc{schneider2026optimal,
title = {{Optimal location of small favourable regions for Robin eigenvalues with indefinite weights}},
author = {Baruch Schneider and Diana Schneiderová and Yifan Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14763},
eprint = {2609.14763},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14763},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Optimal location of small favourable regions for Robin eigenvalues with indefinite weights
AU - Baruch Schneider
AU - Diana Schneiderová
AU - Yifan Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.14763
UR - https://arxiv.org/abs/2609.14763
AB - We consider the positive principal eigenvalue of an elliptic problem with a Robin boundary condition and bang--bang indefinite weight κ1_D−1_Ω∖ D, and ask where a favourable region D of prescribed small volume |D|=δ should be located. Put ε=δ^1/N and τ_δ=α_δε. We prove that there is a finite threshold τ_*=τ_*(N,κ), independent of the ambient domain, such that δ^2/NΛ_δ(α_δ)→Λ_ℍ(τ) whenever τ_δ→τ<∞. If τ<τ_*, optimal small regions concentrate at the boundary. If τ>τ_*, their concentration centres move to distances much larger than ε from the boundary, and after recentring and rescaling the favourable sets converge in measure to the whole-space optimal ball. At τ=τ_*, the half-space problem admits a compact boundary optimiser as well as minimising sequences escaping to infinity. In the finer regime τ_δ=τ_*+σε+o(ε), we determine the first-order competition between the boundary and interior configurations. Tangential symmetry of compact threshold optimisers reduces the geometric correction to mean curvature. In particular, every fixed finite Robin coefficient is asymptotically in the boundary regime. Numerical experiments illustrate the boundary--interior transition, the three cases in the first-order selection law, and the curvature-dependent boundary location; for N=2 and κ=1 they suggest a transition near τ_*≈3.2.
ER -