APPROXIMATION OF GROUND STATE EIGENVALUES AND EIGENFUNCTIONS OF DIRICHLET LAPLACIANS

Abstract

Let Ω be a bounded connected open set in ℝ^N, N[ges ]2, and let −Δ_Ω[ges ]0 be the Dirichlet Laplacian defined in L²(Ω). Let λ_Ω>0 be the smallest eigenvalue of −Δ_Ω, and let ϕ_Ω>0 be its corresponding eigenfunction, normalized by ∥ϕ_Ω∥₂=1. For sufficiently small ε>0 we let R(ε) be a connected open subset of Ω satisfying

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Let −Δ_ε[ges ]0 be the Dirichlet Laplacian on R(ε), and let λ_ε>0 and ϕ_ε>0 be its ground state eigenvalue and ground state eigenfunction, respectively, normalized by ∥ϕ_ε∥₂=1. For functions f defined on Ω, we let S_εf denote the restriction of f to R(ε). For functions g defined on R(ε), we let T_εg be the extension of g to Ω satisfying

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