5 papers
A Priori Log-Concavity Estimates for Dirichlet Eigenfunctions
Gabriel Khan, Soumyajit Saha, Malik Tuerkoen
In this paper, we establish a priori log-concavity estimates for the first Dirichlet eigenfunction of convex domains of a Riemannian manifold. Specifically, we focus on cases where…
Spectral Gap Estimates on Conformally Flat Manifolds
Gabriel Khan, Malik Tuerkoen
The fundamental gap is the difference between the first two Dirichlet eigenvalues of a Schrödinger operator (and the Laplacian, in particular). For horoconvex domains in hyperbolic…
Concavity Properties of Solutions of Elliptic Equations under Conformal Deformations
Gabriel Khan, Soumyajit Saha, Malik Tuerkoen
We study the Dirichlet problem for the weighted Schrödinger operator \[-Δu +Vu = λρu,\] where is a positive weighting function and is a potential. Such equations appear nat…
Modulus of Concavity and Fundamental Gap Estimates on Surfaces
Gabriel Khan, Malik Tuerkoen, Guofang Wei
The fundamental gap of a domain is the difference between the first two eigenvalues of the Laplace operator. In a series of recent and celebrated works, it was shown that for conve…
Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature
Gabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen +1
We study the log-concavity of the first Dirichlet eigenfunction of the Laplacian for convex domains. For positively curved surfaces satisfying a condition involving the curvature a…