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

In this paper we prove that if Ω   and Ω′Ω′ are close enough for the complementary Hausdorff distance and their boundaries satisfy some geometrical and topological conditions then|λ1−λ1′|⩽C|Ω△Ω′|αN where λ1λ1 (resp. λ1′) is the first Dirichlet eigenvalue of the Laplacian in Ω   (resp. Ω′Ω′) and |Ω△Ω′||Ω△Ω′| is the Lebesgue measure of the symmetric difference. Here the constant α<1α<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.