On the lowest eigenvalue of the Laplacian with Neumann boundary condition at a small obstacle

We study the lowest eigenvalue λ 1 ( ε ) of the Laplacian - Δ in a bounded domain Ω ⊂ R d , d ⩾ 2 , from which a small compact set K ε ⊂ B ε has been deleted, imposing Dirichlet boundary conditions along ∂ Ω and Neumann boundary conditions on ∂ K ε . We are mainly interested in results that require minimal regularity of ∂ K ε expressed in terms of a Poincaré condition for the domains Ω ⧹ ε - 1 K ε . We then show that λ 1 ( ε ) converges to Λ 1 , the first Dirichlet eigenvalue of Ω , as ε → 0 . Assuming some more regularity we also obtain asymptotic bounds on λ 1 ( ε ) - Λ 1 , for ε small, where we employ an idea of [Burenkov and Davies, J. Differential Equations 186 (2002) 485–508].