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On Green's function for the Laplace operator in an unbounded domain. Hewgill, Denton Elwood

Abstract

This thesis Investigates the Green's functions for the operator T defined by [ Equation omitted ] Here H ¹₀ (E) is a standard Sobolev space, Δ is the Laplacian, and E is a domain in which is taken to be "quasi-bounded". In particular we assume that E lies in the half-space x₁ > 0 and is bounded by the surface obtained by rotating φ(x₁) about the x₁-axis, where φ is continuous, φ(x₁) > 0 and φᵏ∈ L₁(0,+∞) for some k > 0. The Green's function G(x,y,λ) for the operator T + λ is obtained as the limit of the Green's functions for the well known problem on the truncated domain Eₓ=E ∩ [X₁ < X]. Most of the expected properties of the function are developed including the iii equality [ Equation omitted ] where K is the fundamental singularity for the domain. The eigenvalues and eigenfunctions are constructed, and it is shown that [ Equation omitted ] where λₓ,n and λn are the eigenvalues for the problem on Eₓ and E respectively. Furthermore, it is shown that the eigenvalues {λn} are positive with no finite limit point, and the corresponding eigenfurictions are complete. A detailed calculation involving the inequality displayed above shows that some iterate (Gᵏ ̊) of G(x,y,λ) is a Hilbert-Schmidt kernel. From this property of Gᵏ ̊ it follows that the series ∑λn ˉ²ᵏ ̊ is convergent. From the convergence of this series three results are derived. The first one is an expansion formula in terms of the complete set of eigenfunctions, and the second is that some iterate of the Green's function tends to zero on the boundary. The last one Is the construction of the solution H(x,λ,f), for the boundary value problem ΔH + λH = f [ Equation omitted ] for a sufficiently regular f on E. The final property of the Green's function, namely, that G(x,y,λ) tends to zero on the boundary, is proved using the fact that Gᵏ ̊is zero on the boundary, and certain inequaiitites estimating the iterates G(x,y, λ) is also shown to be unique. The asymptotic formula [ Equation omitted ] a generalization of the usual asymptotic formula of Weyl for the eigenvalues, first given by C. Clark, is derived for these quasi-bounded domains. Finally, the usual asymptotic formula due to Carleman for the eigenfunctions is shown to remain valid.