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Maximization of the second non-trivial Neumann eigenvalue
Acta Mathematica ( IF 3.7 ) Pub Date : 2019-01-01 , DOI: 10.4310/acta.2019.v222.n2.a2
Dorin Bucur 1 , Antoine Henrot 2
Affiliation  

In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of R N with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2. As a consequence, the P{\'o}lya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.

中文翻译:

第二个非平凡 Neumann 特征值的最大化

在本文中,我们证明了在具有指定测度 m 的 RN 的光滑域上的拉普拉斯算子的第二个(非平凡)诺依曼特征值在两个不相交的测度为 m 2 的球的并集上达到其最大值。因此,P{对于第二个特征值和任意域,Neumann 特征值的 \'o}lya 猜想成立。此外,我们证明了相同不等式的松弛形式在非光滑域和密度的背景下成立。
更新日期:2019-01-01
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