论文标题
$ p $ -laplacian的第一个罗宾特征值
First Robin Eigenvalue of the $p$-Laplacian on Riemannian Manifolds
论文作者
论文摘要
我们考虑第一个Robin Eigenvalue $ ol_p（m，\ a）$，用于$ p $ -laplacian在紧凑的Riemannian歧管$ m $带有非空平滑边界上的$ p $ -laplacian，其中$ \ a \ in \ in \ r $是robin参数。首先，我们证明Cheng Type的特征值比较定理，以$ _p（m，\ a）$。其次，当$ \ a> 0 $时，我们在尺寸（m，\ a）$的尺寸（inradius）中建立了较锋利的下限，inradius，ricci曲率下限和边界平均值曲率下限，通过与相关的一维特征值问题进行比较。 $ \ a <0 $时，下限将变成上限。我们的结果涵盖了$ p $ -laplacian的第一个dirichlet特征值的相应比较定理。
We consider the first Robin eigenvalue $ł_p(M,\a)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty smooth boundary, with $\a \in \R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for $ł_p(M,\a)$. Secondly, when $\a>0$ we establish sharp lower bound of $ł_p(M,\a)$ in terms of dimension, inradius, Ricci curvature lower bound and boundary mean curvature lower bound, via comparison with an associated one-dimensional eigenvalue problem. The lower bound becomes an upper bound when $\a<0$. Our results cover corresponding comparison theorems for the first Dirichlet eigenvalue of the $p$-Laplacian when letting $\a \to +\infty$.
