具有小/大扩散系数的二维线性二阶椭圆算子的主特征值的渐近行为

Asymptotic Behavior of the Principal Eigenvalues of Two-Dimensional Linear Second-Order Elliptic Operators with Small/Large Diffusion Coefficients

摘要: 本文研究一类带有强/弱扩散和对流项的二维线性二阶椭圆算子在Neumann边界条件下的主特征值问题：$-D\Delta \phi -2\alpha \nabla m\left(x\right)\cdot \nabla \phi +V\left(x\right)\phi =\lambda \phi$ 。我们重点分析主特征值在扩散系数趋于零或对流强度趋于无穷时的渐近行为，特别关注特征函数的集中位置、局域结构及其与几何特征之间的关系。为此，本文在奇异曲线邻域引入适应局部几何的移动坐标系，并结合一致渐近展开、局部曲率及势函数的精细估计，系统建立了主特征函数在极限情形下出现的结构以及主特征值的显式渐近式。所得结果揭示了扩散、对流与几何曲率之间的潜在耦合机制，为理解参数依赖型椭圆算子的谱性质提供了统一框架，并对已有文献中的相关结论进行了自然推广。

Abstract: This paper investigates the principal eigenvalue problem for a class of two-dimensional linear second-order elliptic operators with strong/weak diffusion and convection terms under Neumann boundary conditions：$-D\Delta \phi -2\alpha \nabla m\left(x\right)\cdot \nabla \phi +V\left(x\right)\phi =\lambda \phi$ . We focus on analyzing the asymptotic behavior of the principal eigenvalues as the diffusion coefficient approaches zero or the convection intensity tends to infinity, paying particular attention to the localization of eigenfunctions, their local structure, and their relationship with geometric parameters. To this end, we introduce a moving coordinate system adapted to the local geometry in the vicinity of singular curves. By combining uniform asymptotic expansions, local curvature estimates, and refined estimates for the potential function, we systematically establish the structure of the principal eigenfunction in the limit cases and derive explicit asymptotic formulas for the principal eigenvalues. The results reveal underlying coupling mechanisms among diffusion, convection, and geometric curvature, providing a unified framework for understanding the spectral properties of parameter-dependent elliptic operators and offering a natural extension of related conclusions in the existing literature.