Perturbing the principal Dirichlet eigenfunction

2025/04/26 by Chao, Brian, Saloff-Coste, Laurent · 1 citation
#31C25 #35B51 #35K08 (Secondary) #60J60 (Primary) 35J25 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2504.18783

Abstract

We study the principal Dirichlet eigenfunction φ_U when the domain U is a perturbation of a bounded inner uniform domain in a strictly local regular Dirichlet space. We prove that if U is suitably contained in between two inner uniform domains, then φ_U admits two-sided bounds in terms of the principal Dirichlet eigenfunctions of the two approximating domains. The main ingredients of our proof include domain monotonicity properties associated to Dirichlet boundary conditions, intrinsic ultracontractivity estimates, and parabolic Harnack inequality. As an application of our results, we give explicit expressions comparable to φ_U for certain domains U⊆ ℝⁿ, as well as improved Dirichlet heat kernel estimates for such domains. We also prove that under a uniform exterior ball condition on U, a point achieving the maximum of φ_U is separated away from the boundary, complementing a result of Rachh and Steinerberger arXiv:1608.06604. Our principal Dirichlet eigenfunction estimates are applicable to second-order uniformly elliptic operators in Euclidean space, Riemannian manifolds with nonnegative Ricci curvature, and Lie groups of polynomial volume growth.

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