Upper bound for the first nonzero eigenvalue related to the p-Laplacian

Abstract

Let M denote a complete, simply connected Riemannian manifold with sectional curvature K M ≤ k and Ricci curvature Ric M ≥ (n − 1)K, where k, K ∈ R. Then for a bounded domain Ω ⊂ M with smooth boundary, we prove that the first nonzero Neumann eigenvalue µ 1 (Ω) ≤ Cµ 1 (B k (R)). Here B k (R) is a geodesic ball of radius R > 0 in the simply connected space form M k such that vol(Ω) = vol(B k (R)), and C is a constant which depends on the volume, diameter of Ω and the dimension of M.

REILLY-TYPE UPPER BOUNDS FOR THE p-STEKLOV PROBLEM ON SUBMANIFOLDS

“…Recently, Verma obtained upper bounds for the first eigenvalue σ 1,p of the p-Steklov problem (S p ) for Euclidean domains [21]. She proved that for a bounded domain Ω with smooth boundary,…”

EXTRINSIC UPPER BOUNDS THE FIRST EIGENVALUE OF THE p-STEKLOV PROBLEM ON SUBMANIFOLDS

“…In a very recent paper, V. Sheela [13] obtain upper bound for the first eigenvalue of the p-Steklov problem (S) for Euclidean domain. Namely she proves that for a bounded domain Ω with smooth boundary, then λ 1 1 R p−1 (resp.…”