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

We study the lowest eigenvalue λ1(ε)λ1(ε) of the Laplacian -Δ-Δ in a bounded domain Ω⊂RdΩ⊂Rd, d⩾2d⩾2, from which a small compact set Kε⊂BεKε⊂Bε has been deleted, imposing Dirichlet boundary conditions along ∂Ω∂Ω and Neumann boundary conditions on ∂Kε∂Kε. We are mainly interested in results that require minimal regularity of ∂Kε∂Kε expressed in terms of a Poincaré condition for the domains Ω⧹ε-1KεΩ⧹ε-1Kε. We then show that λ1(ε)λ1(ε) converges to Λ1Λ1, the first Dirichlet eigenvalue of ΩΩ, as ε→0ε→0. Assuming some more regularity we also obtain asymptotic bounds on λ1(ε)-Λ1λ1(ε)-Λ1, for εε small, where we employ an idea of [Burenkov and Davies, J. Differential Equations 186 (2002) 485–508].