Approximation of Ground State Eigenvalues and Eigenfunctions of Dirichlet Laplacians

Let Ω be a bounded connected open set in RN, N ⩾ 2, and let −ΔΩ⩾0 be the Dirichlet Laplacian defined in L2(Ω). Let λΩ > 0 be the smallest eigenvalue of −ΔΩ, and let φΩ > 0 be its corresponding eigenfunction, normalized by ∥φΩ∥2 = 1. For sufficiently small ε>0 we let R(ε) be a connected open subset of Ω satisfying R ( ε ) ⊇ { x ∈ Ω : dist ( x , ∂ Ω ) ⩾ ε } . Let −Δε ⩾ 0 be the Dirichlet Laplacian on R(ε), and let λε>0 and φε>0 be its ground state eigenvalue and ground state eigenfunction, respectively, normalized by ∥φε∥2=1. For functions f defined on Ω, we let Sεf denote the restriction of f to R(ε). For functions g defined on R(ε), we let Tεg be the extension of g to Ω satisfying ( T ε g ) ( z ) = { g ( z ) if z ∈ R ( ε ) , 0 if z ∈ Ω ∖ R ( ε ) . 1991 Mathematics Subject Classification 47F05.