Universal inequalities of the poly-drifting Laplacian on the Gaussian and cylinder shrinking solitons
type: Item
instance of: scholarly article
title: Universal inequalities of the poly-drifting Laplacian on the Gaussian and cylinder shrinking solitons
zbMATH Open document ID: 1325.35121
author: Feng Du
author: Jing Mao
author: Qiaoling Wang
author: Chuanxi Wu
published in: Annals of Global Analysis and Geometry
publication date: 2015-10-02T00:00:00Z
review text: Let \(\Omega\) be a bounded domain inside a complete Riemannian manifold \((M,g)\). Let \(L_\phi:=\Delta-(\nabla\phi,\nabla\cdot)\) be the weighted Laplacian or the drifting Laplacian, where \(\phi\in C^\infty(M)\). Impose boundary conditions defined by the boundary operator \(\mathcal{B}u=\{u\oplus\partial_\nu u\oplus\dots\oplus\partial_\nu^{m-1}\}|_{\partial M}\), where \(\nu\) is the inward unit normal vector field. The operator \((-L_\phi)^m\) is self-adjoint with these boundary conditions with respect to the inner-product defined by the conformally equivalent measure \(e^{-\phi}\text{dvol}_g\). The authors consider the eigenvalue problem \((-L_\phi)^mu=\Lambda u\) in \(\Omega\) with these boundary conditions and obtain a general inequality for lower-order eigenvalues. They obtain some universal inequalities for lower-order eigenvalues in the case when the ambient space is Euclidean space or the unit sphere. They also obtain universal inequalities in the Gaussian and cylinder shrinking solitons. The first section is an introduction to the problem, the second section treats some preliminaries, and the third section establishes the main results.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 6488356
zbMATH Keywords: eigenvalues
zbMATH Keywords: universal inequalities
zbMATH Keywords: poly-drifting Laplacian
zbMATH Keywords: Gaussian shrinking soliton
zbMATH Keywords: cylinder shrinking soliton
MaRDI profile type: Publication
full work available at URL: s10455-015-9469-x
OpenAlex ID: W302085301
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DOI: 10.1007/S10455-015-9469-X
description: scientific article; zbMATH DE number 6488356