发表机构
Università San Raffaele Roma（罗马圣拉法埃勒大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究锥体内拉普拉斯第一特征值的极小化问题，证明球扇区为临界形状，并给出稳定性条件，进而证明极小元存在且满足超定问题。
AI 中文摘要
本文研究了锥体内区域上拉普拉斯算子第一特征函数的相对超定问题，以及给定测度集合中最小化第一特征值的相关问题。通过形状导数分析，我们证明了球扇区是临界形状，并获得了锥体上其稳定性/不稳定性的几何条件。通过集中紧性论证，我们证明了极小元的存在性，且该极小元是有界、开、连通的，其相对边界几乎处处正则。通过另一种区域变分论证，我们得出结论：极小元承认超定问题的解。
英文摘要
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.
Comments38 pages. All comments are much welcome
