Abstract
In this paper, we firstly study the eigenvalue problem of a systemof elliptic equations with drift and get some universal inequalities of PayneP′olya-Weinberger-Yang type on a bounded domain in Euclidean spaces and inGaussian shrinking solitons. Furthermore, we study two kinds of the clampedplate problems and the buckling problems for the bi-drifting Laplacian and getsome sharp lower bounds for the first eigenvalue for these eigenvalue problemon compact manifolds with boundary and positive m-weighted Ricci curvatureor on compact manifolds with boundary under some condition on the weightedRicci curvature.
Highlights
For a given complete n-dimensional Riemannian manifold (M, ) with a metric, the triple (M, e−f dν) is called a smooth metric measure space, where f is a smooth real-valued function on M and dν is the Riemannian volume element related to
Let Ω be a bounded domain in an n-dimensional Euclidean space Rn, in this paper, we will consider the following eigenvalue problem:
We know that a metric measure space is not necessarily compact when Ricf ≥ λ > 0, unlike in the case of Riemannian manifolds where such a complete one is compact if its Ricci curvature is bounded from below uniformly by some positive constant
Summary
For a given complete n-dimensional Riemannian manifold (M, , ) with a metric , , the triple (M, , , e−f dν) is called a smooth metric measure space, where f is a smooth real-valued function on M and dν is the Riemannian volume element related to , (sometimes, we call dν the volume density). On a smooth metric measure space (M, , , e−f dν), we can define the so-called drifting Laplacian ( called weighted Laplacian) ∆f as follows. Let Ω be a bounded domain in an n-dimensional Euclidean space Rn, in this paper, we will consider the following eigenvalue problem:. Eigenvalue, universal inequalities, system of elliptic equations with drift, drifting Laplacian, m-weighted Ricci curvature, Gaussian shrinking soliton. On smooth metric measure spaces, we can define the so-called ∞-BakryEmery Ricci tensor Ricf by. We would like to give an important example of Ricci solitons From this example, we know that a metric measure space is not necessarily compact when Ricf ≥ λ > 0, unlike in the case of Riemannian manifolds where such a complete one is compact if its Ricci curvature is bounded from below uniformly by some positive constant
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
