带漂移的协作椭圆系统主特征值的奇异性分析
Singular analysis of the principal eigenvalue for cooperative elliptic systems with drifts
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中文总结 AI 辅助
本文研究带漂移的一维协作椭圆系统主特征值的小扩散极限，证明其由Aubry集分解出的各类局部谱和输运问题的最小值决定，并给出对数特征函数轮廓。
中文摘要 AI 辅助
我们确定了带分量相关漂移和Neumann边界条件的一维协作椭圆系统主特征值的小扩散极限。对于标量方程，该极限由漂移平衡点和边界点处的局部数据决定。我们证明这种局部化性质对系统失效：分量间的切换可使某些区间在谱上具有相关性。我们首先证明所有分量特征函数的对数变换收敛到同一个函数，该函数求解一个标量Hamilton-Jacobi方程。相关的Aubry集决定了主特征值的渐近行为。在非退化假设下，该集合分解为孤立点、正则区间和包含公共漂移零点的复合区间。然后我们通过局部Ornstein-Uhlenbeck谱和区间输运问题确定每一类的有效值，并建立主特征值的极限为这些类值的最小值。唯一的最小化类也决定了共同的对数特征函数轮廓。
英文摘要
We determine the small-diffusion limit of the principal eigenvalue for a one-dimensional cooperative elliptic system with component-dependent drifts and Neumann boundary conditions. For scalar equations, this limit is determined by local data at the drift equilibria and boundary points. We show that such localization fails for systems: switching between components can make some intervals spectrally relevant. We first prove that the logarithmic transforms of all component eigenfunctions converge to the same function, which solves a scalar Hamilton-Jacobi equation. The associated Aubry set determines the asymptotic behavior of the principal eigenvalue. Under nondegeneracy assumptions, this set decomposes into isolated points, regular intervals, and composite intervals containing common drift zeros. We then identify the effective value of each class through local Ornstein-Uhlenbeck spectra and interval transport problems, and establish that the limit of the principal eigenvalue is the minimum of these class values. A unique minimizing class also determines the common logarithmic eigenfunction profile.
发表机构
- School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
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