Let Ω be a bounded connected open set in ℝN, N[ges ]2, and let −ΔΩ[ges ]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

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Let −Δε[ges ]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

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