向量$p$ -Laplacian周期特征值的结构
On the Structure of Periodic Eigenvalues of the Vectorial $p$-Laplacian
摘要：
在本文中，我们将解决Manásevich和Mawhin二十年前提出的一个公开问题的结构的周期特征值的向量$p$ -拉普拉斯算子。这是一个平面上的欧拉-拉格朗日方程或高维欧几里得空间中的欧拉-拉格朗日方程。得到的主要结果是：对于除$2$以外的任何指数$p$，平面上的向量$p$ -Laplacian将容许无穷多个具有给定周期的不同周期特征值序列.这些特征值序列是用我们将要介绍的标度动量的概念构造的。整个证明是基于完全可积性的等效哈密顿系统，巧妙的减少到$2$维动力系统，和数论区分不同序列的特征值。对新的特征值和特征函数序列进行了数值模拟。对光谱集的全景将强加几个进一步的解释。
英文摘要：
In this paper we will solve an open problem raised by Manásevich and Mawhin twenty years ago on the structure of the periodic eigenvalues of the vectorial $p$-Laplacian. This is an Euler-Lagrangian equation on the plane or in higher dimensional Euclidean spaces. The main result obtained is that for any exponent $p$ other than $2$, the vectorial $p$-Laplacian on the plane will admit infinitely many different sequences of periodic eigenvalues with a given period. These sequences of eigenvalues are constructed using the notion of scaling momenta we will introduce. The whole proof is based on the complete integrability of the equivalent Hamiltonian system, the tricky reduction to $2$-dimensional dynamical systems, and a number-theoretical distinguishing between different sequences of eigenvalues. Some numerical simulations to the new sequences of eigenvalues and eigenfunctions will be given. Several further conjectures towards to the panorama of the spectral sets will be imposed.
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