1 Completeness of a Set of Eigenfunctions 1.1 The Neumann Boundary Condition

In this lecture, we begin examining a generalized look at the Laplacian Eigenvalue Problem, particularly related to generalized domains. Our goal here is to establish the notion of the completeness of a set of eigenfunctions, which we then use to justify separation of variables, a tool we so far have taken for granted. Basic references for this lecture are [1, Sec. 11.3, 11.5], [2, Chap. 11] and [3, Sec. 3.3]. For more advanced treatments, see [4, Sec. 6.5]. In the earlier lectures, we used:{ νj or λ (N) j for the Neumann-Laplacian eigenvalues, ψj or ϕ (N) j for the Neumann-Laplacian eigenfunctions. For simplicity, let us use (νj, ψj) for Neumann BC and (λj, ϕj) for Dirichlet BC. 1 And we number νj in ascending order: 0 = ν1 < ν2 ≤ ν3 ≤ · · · , also recall that ψ1(x) = const. Theorem 1.1. For the Neumann Condition, we define a “trial function ” as any w(x) ∈ C2(Ω) such that w(x) 6 ≡ 0. Then similarly to the Dirichlet case, (MP), (MP)n, RRA (see Lecture 9), and the minimax principle are all valid. Note: w ∈ C2(Ω) means that there is no constraint at ∂Ω. In particular, w may not satisfy the Neumann condition (nor the Dirichlet condition). As such, the Neu-mann condition is also referred to as the free condition and the Dirichlet condition is referred to as the fixed condition. Proof. Define m = min w∈C2(Ω) w 6≡0 {‖∇w‖2 ‖w‖ Let u ∈ C2(Ω) attain this minimum. Set w = u + εv, ∀ v ∈ C2(Ω). The similar procedure leads to (as a result of variational calculus):∫ Ω (−∇u · ∇v +muv) dx = 0, ∀ v ∈ C2(Ω) (1) By Green’s first identity, (1) ⇐