PreprintCC BY 4.0arXiv
Non-radial minimizers for the first eigenvalue of the Laplacian in cones and a related overdetermined problem
Abstract
In this work, we consider relative overdetermined problems for the first eigenfunction of the Laplacian for domains in cones, and the related question of minimizing the first eigenvalue among sets of a given fixed measure. By means of a shape derivative analysis, we show that the spherical sector is a critical shape and obtain a geometric condition on the cone for its stability/instability. By a concentration-compactness argument, we prove the existence of a minimizer, which moreover is bounded, open, connected, and whose relative boundary is regular almost everywhere. By another domain variation argument, we conclude that the minimizers admit a solution for the overdetermined problem.
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BibTeX
@misc{afonso2026nonradial,
title = {{Non-radial minimizers for the first eigenvalue of the Laplacian in cones and a related overdetermined problem}},
author = {Danilo Gregorin Afonso},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12879},
eprint = {2609.12879},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12879},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-radial minimizers for the first eigenvalue of the Laplacian in cones and a related overdetermined problem
AU - Danilo Gregorin Afonso
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12879
UR - https://arxiv.org/abs/2609.12879
AB - In this work, we consider relative overdetermined problems for the first eigenfunction of the Laplacian for domains in cones, and the related question of minimizing the first eigenvalue among sets of a given fixed measure. By means of a shape derivative analysis, we show that the spherical sector is a critical shape and obtain a geometric condition on the cone for its stability/instability. By a concentration-compactness argument, we prove the existence of a minimizer, which moreover is bounded, open, connected, and whose relative boundary is regular almost everywhere. By another domain variation argument, we conclude that the minimizers admit a solution for the overdetermined problem.
ER -