论文标题
$ p $ laplacian的特征值的共形上限
Conformal upper bounds for the eigenvalues of the $p$-Laplacian
论文作者
论文摘要
在本说明中，我们介绍了$ p $ laplacian在完全$ n $ dimensional-demensional riemannian歧管和Neumann边界条件以及紧凑型（无边界）Riemannian歧管上的平滑域上的差异特征值的上限。特别是，我们以$ 1 <p \ leq n $的给定歧管$（m，g）$的共形类别提供上限，当我们修复公制$ g $时，所有$ p> 1 $的上限。为此，我们使用公制方法来构建合适的测试函数，以进行特征值的变异表征。上限与由于弗里德兰德引起的特征值的著名渐近估计。我们还以等速度比率的riemannian歧管中的高空曲面上的变异特征值提出了上限。
In this note we present upper bounds for the variational eigenvalues of the $p$-Laplacian on smooth domains of complete $n$-dimensional Riemannian manifolds and Neumann boundary conditions, and on compact (boundaryless) Riemannian manifolds. In particular, we provide upper bounds in the conformal class of a given manifold $(M,g)$ for $1<p\leq n$, and upper bounds for all $p>1$ when we fix a metric $g$. To do so, we use a metric approach for the construction of suitable test functions for the variational characterization of the eigenvalues. The upper bounds agree with the well-known asymptotic estimate of the eigenvalues due to Friedlander. We also present upper bounds for the variational eigenvalues on hypersurfaces bounding smooth domains in a Riemannian manifold in terms of the isoperimetric ratio.
