Eigenvalue estimates for a class of elliptic differential operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

Abstract

In this paper, we obtain eigenvalue estimates for a larger class of elliptic differential operators in divergence form on a bounded domain in a complete Riemannian manifold isometrically immersed in Euclidean space. As an application, we give eigenvalue estimates in the Gaussian shrinking soliton, and we find a domain that makes the behavior of these estimates similar to the estimates for the case of the Laplacian. Moreover, we also give an answer to the generalized conjecture of Pólya.

Final version which has been accepted for publication in Zeitschrift für Angewandte Mathematik und Physik

References in corpus (2)

• Estimates of eigenvalues of an elliptic differential system in divergence form
• Inequalities for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds