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

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 | λ 1 − λ 1 ′ | ⩽ C | Ω △ Ω ′ | α N where λ 1 (resp. λ 1 ′ ) 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.