Quantitative stability for the first Dirichlet eigenvalue in Reifenberg flat domains in RN

Abstract

In this paper we prove that if Ω and Ω′ are close enough for the complementary Hausdorff distance and their boundaries satisfy some geometrical and topological conditions then| λ1 - λ1′ | ≤ C | Ω △ Ω′ |frac(α, N) where λ1 (resp. λ1′) is the first Dirichlet eigenvalue of the Laplacian in Ω (resp. Ω′) and | Ω △ Ω′ | is the Lebesgue measure of the symmetric difference. Here the constant α < 1 could be taken arbitrary close to 1 (but strictly less) and C is a constant depending on a lot of parameters including α, dimension N and some geometric properties of the domains. © 2009 Elsevier Inc. All rights reserved.