Eigenvalue inequalities of elliptic operators in weighted divergence form on smooth metric measure spaces

Yuming Zhu,Feng Du,Gusheng Liu

doi:10.1186/s13660-016-1130-0

Yuming Zhu, Feng Du + Show 1 more

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https://doi.org/10.1186/s13660-016-1130-0

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Abstract

In this paper, we study the eigenvalue problem of elliptic operators in weighted divergence form on smooth metric measure spaces. First of all, we give a general inequality for eigenvalues of the eigenvalue problem of elliptic operators in weighted divergence form on compact smooth metric measure space with boundary (possibly empty). Then applying this general inequality, we get some universal inequalities of Payne-Polya-Weinberger-Yang type for the eigenvalues of elliptic operators in weighted divergence form on a connected bounded domain in the smooth metric measure spaces, the Gaussian shrinking solitons, and the general product solitons, respectively.

Highlights

1 Introduction A smooth metric measure space is a Riemannian manifold equipped with some measure which is absolutely continuous with respect to the usual Riemannian measure
Let be a bounded domain in a smooth metric measure space (M, e–f dν), and let A : → End(T ) be a smooth symmetric and positive definite section of the bundle of all endomorphisms of T, we can define the elliptic operator in weighted divergence form as
When f is a constant, Lf becomes the elliptic operator in divergence form, for some recent developments about universal inequalities of the eigenvalue of elliptic operator in divergence form

Summary

Introduction

A smooth metric measure space is a Riemannian manifold equipped with some measure which is absolutely continuous with respect to the usual Riemannian measure. It is a natural problem how to get the universal inequalities of the eigenvalues of elliptic operator in weighted divergence form. For the fourth-order elliptic operator in weighted divergence, we can consider the following eigenvalue problem:. On smooth metric measure spaces, we can define the so-called weighted Ricci curvature Ricf given by. We will give some universal inequalities for the Dirichlet eigenvalues in a connected bounded domain on the Gaussian shrinking solitons and general product solitons. For the Dirichlet problem of the operator L, some universal inequalities have been obtained by Cheng and Peng [ ] In this case, our results can be regarded as conclusions for the Dirichlet problem of the elliptic operator in weighted divergence form

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