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 metric for $1<p⩽n$, and upper bounds for all $p>1$ when we fix a metric. 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.

中文翻译：

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p-拉普拉斯算子特征值的共形上限

在本说明中，我们提出了变分特征值的上限$p$- 完全光滑域上的拉普拉斯算子$n$维黎曼流形和纽曼边界条件，以及紧凑（无边界）黎曼流形。特别是，我们为给定度量的保形类提供了上限$1<p⩽n$, 和所有的上限$p>1$当我们修复一个指标时。为此，我们使用度量方法为特征值的变分特征构建合适的测试函数。上界与著名的弗里德兰德对特征值的渐近估计一致。我们还根据等周比给出了黎曼流形中边界平滑域的超曲面上的变分特征值的上限。