Lower bound estimates for the first eigenvalue of the weighted p-Laplacian on smooth metric measure spaces

Abstract

New lower bounds of the first nonzero eigenvalue of the weighted p-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the m-Bakry-Émery Ricci curvature, the Escober-Lichnerowicz-Reilly type estimates are proved; under the assumption of nonnegative ∞-Bakry-Émery Ricci curvature and the m-Bakry-Émery Ricci curvature bounded from below by a non-positive constant, the Li-Yau type lower bound estimates are given. The weighted p-Bochner formula and the weighted p-Reilly formula are derived as the key tools for the establishment of the above results. Mathematics Subject Classification (2010). Primary 58J05, 58J50; Secondary 35J92.

Weighted Cheeger constant and first eigenvalue lower bound estimates on smooth metric measure spaces

“…This result is a generalization of [17,38] in the case of bounded domains with smooth boundary in Riemannian manifolds. See also similar results [26,35,36,39] for the p-Laplacian or drifting Laplacian eigenvalue problems. For applications, following closely the idea of Hessian comparison estimates applied in [11,19,25] on different problems, we discuss lower bound estimates for the first eigenvalues on the weighted geodesic ball whose radius does not exceed the injectivity radius and submanifolds having bounded weighted mean curvature.…”

“…Inspired by this, Wang and Li [36] combined weighted p-Bochner and p-Reilly formulas with gradient estimate technique to derive lower bound of Escobar-Lichnerowicz-Reilly type on the first eigenvalue of drifting p-Laplacian on a compact smooth metric measure space M in terms of the sign of the Bakry-Émery Ricci curvature. Namely, for p ≥ 2, Ric m φ ≥ kg, k > 0, and the first eigenvalue λ ∂M 1,p of -L φ,p on M, we have Recall that the drifting Laplacian is an important self-adjoint elliptic operator associated with the Ricci soliton from the Ricci flow theory and self-shrinker from the mean curvature flow theory.…”

Monotonicity formulas for the first eigenvalue of the weighted p-Laplacian under the Ricci-harmonic flow

“…Firstly, we introduce the linearized operator of the weighted p-Laplacian on function h ∈ C ∞ (M) defined pointwise at the points ∇h = 0 [23] L φ (f ) := e φ div e -φ |∇h| p-2 G(∇f )…”

On the eigenvalue problem involving the weighted p-Laplacian in radially symmetric domains

“…22), (4.25) and (4.26) there exist C > 0 and µ > 0, both independent of u, such that (4.2) holds. This completes the proof of part (i).…”