AI 中文总结
该研究针对周期格上非对称二阶差分算子的主特征值，建立变分公式与重排不等式，推导相关校正子性质，在一维情形下得到可最大化倾斜主特征值的钟形循环重排。
AI 中文摘要
本文针对周期格上的非对称二阶差分算子的主特征值，建立了变分公式与一个重排不等式。对于一般不可约近邻算子，正右、左特征向量给出了显式鞍点，因此在正剖面与周期对数校正子上，最小-最大值与最大-最小值公式相等；校正子唯一且满足非线性离散通量守恒律。对于指数倾斜的对称扩散，该公式将离散狄利克雷项与势项从精确的非线性周期校正中分离出来。在一维情形下，通量为常数，初等标量通量表示给出了钟形循环重排，该重排可对每一个倾斜量最大化倾斜主特征值。
英文摘要
In this paper, we establish variational formulas and a rearrangement inequality for principal eigenvalues of asymmetric second-order difference operators on periodic lattices. For a general irreducible nearest-neighbor operator, positive right and left eigenvectors give an explicit saddle point and hence equal minimum--maximum and maximum--minimum formulas over positive profiles and periodic logarithmic correctors. The corrector is unique and satisfies a nonlinear discrete flux-conservation law. For exponentially tilted symmetric diffusion, the formula separates the discrete Dirichlet and potential terms from an exact nonlinear periodic correction. In one dimension the flux is constant, and an elementary scalar-flux representation yields a bell-shaped cyclic rearrangement that maximizes the tilted principal eigenvalue for every tilt.
