AI 中文总结
本文研究平面域几何与受均匀磁场和Dirichlet边界条件约束的非线性$p$-Laplacian谱的关系，建立了全平面谱底的下界（推广Landau能级估计）及凸域第一特征值的Pólya型渐近锐利界，并提出了开放问题。
AI 中文摘要
我们研究了平面域几何与非线性$p$-Laplacian谱之间的相互作用，该算子受到均匀磁场和Dirichlet边界条件约束。首先，我们在整个平面上建立了谱底的下界，将经典的Landau能级估计推广到非线性情形。其次，我们证明了有界凸平面域上第一特征值的Pólya型界，该界以面积、周长、最小宽度和磁场强度表示，并表明这些界沿合适的薄化域序列是渐近锐利的。最后，我们提出了一些带有开放问题的评论。
英文摘要
We investigate the interplay between the geometry of a planar domain and the spectrum of the nonlinear $p$-Laplacian, subject to a homogeneous magnetic field and Dirichlet boundary conditions. First, we establish a lower bound for the bottom of the spectrum on the whole plane, extending the classical Landau-level estimate to the nonlinear regime. Second, we prove Pólya-type bounds for the first eigenvalue on bounded convex planar domains in terms of the area, perimeter, minimal width and magnetic field strength, and show that they are asymptotically sharp along sequences of suitable thinning domains. Finally, we post some remarks with open problems.
