第二个非平凡Neumann特征值的最大化

IF 4.9 1区数学 Q1 MATHEMATICS

Acta Mathematica Pub Date : 2018-01-23 DOI:10.4310/ACTA.2019.V222.N2.A2

D. Bucur, A. Henrot

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引用次数: 30

摘要

本文证明了在给定测度为m的rn光滑域上拉普拉斯算子的第二(非平凡)Neumann特征值在两个测度为m2的不相交球的并集上达到最大值。因此，诺伊曼特征值的P{\'o}lya猜想对第二个特征值和任意域都成立。此外，我们还证明了在非光滑域和非光滑密度的情况下，相同不等式的松弛形式成立。

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Maximization of the second non-trivial Neumann eigenvalue

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.

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