Preprint

Inserted: 5 oct 2026
Last Updated: 5 oct 2026

Year: 2026
Doi: https://doi.org/10.48550/arXiv.2609.12879
Links: https://arxiv.org/abs/2609.12879

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 stabilityinstability. 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.