arXiv subjects

Zhengchao Ji

Publications and source records attributed to Zhengchao Ji.

3 recordsLinked to original sources

arXiv2026

Improved lower bounds for Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded Euclidean domains

In this paper, we establish Brezin-Li-Yau type lower bounds for averaged sums of Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded domains in Euclidean spaces. By deriving expansions of two binary polynomials which may be of independent interest, we improve several existing lower bounds of this kind in the literature. Furthermore, our lower bounds are optimal in the sense that our expansions capture all positive terms, whereas previous works only provided certain lower bounds for these two binary polynomials, effectively capturing only a subset of the positive terms identified in our expansions.

math.AP↗

arXiv2025

Deep estimates for higher eigenvalues of the poly-Laplacian

We investigate the lower bound for higher eigenvalues $λ_i$ of the poly-Laplace operator on a bounded domain and improve the famous Li-Yau inequality and its related results. Firstly, we consider the low dimensional cases for the Pólya conjecture, the clamped plate problem and the eigenvalue problem of the poly-Laplacian and deliver a series of deep eigenvalue inequalities for these problems respectively. Secondly, we establish a sharp lower bound for the eigenvalues of the poly-Laplacia in arbitrary dimension under some certain restrictive conditions. Finally, we provide an improved inequality for $λ_i$ in arbitrary dimension without any restrictive conditions. Our results also yield the improvement of the lower bounds for the Stokes eigenvalue problems and the Generalized Pólya conjecture.

math.DG↗

arXiv2020

Lower bounds for eigenvalues of Laplacian operator and the clamped plate problem

In this paper, we give some lower bounds for several eigenvalues. Firstly, we investigate the eigenvalues $λ_i$ of the Laplace operator and prove a sharp lower bound. Moreover, we extent this estimate of the eigenvalues to general cases. Secondly, we study the eigenvalues $Γ_i$ for the clamped plate problem and deliver a sharp bound for the clamped plate problem for arbitrary dimension.

math.DG↗