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
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
Sign-in/Register to view highlights & summary 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 callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
