Eigenvalue estimates for a class of elliptic differential operators in divergence form. (November 2018)

• Record Type: Journal Article
• Authors: Gomes, José. N.V.; Miranda, Juliana F.R.
• Is Part Of: Nonlinear analysis. Volume 176(2018)
• Page Start: 1
• Page End: 19
• Publication Date: 2018-11
• Subjects: primary 35P15 -- secondary 53C42 58J50; Drifting Laplacian -- Eigenvalues -- Elliptic operator -- Immersions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.06.006
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based upon the Weyl's asymptotic formula. As an application, we find a lower bound for the mean of the first k eigenvalues of the drifting Laplacian. In particular, we have extended for this operator a partial solution given by Cheng and Yang for the generalized conjecture of Pólya. We also derive a second-Yang type inequality due to Chen and Cheng, and other two inequalities which generalize results by Cheng and Yang obtained for a domain in the unit sphere and for a domain in the projective space.