STEKLOV REPRESENTATIONS OF GREEN’S FUNCTIONS FOR LAPLACIAN BOUNDARY VALUE PROBLEMS. GILES AUCHMUTY Abstract. This paper describes different representations for solution operators of Lapla- cian boundary value problems on bounded regions in R N ,N ≥ 2 and in exterior regions when N = 3. Null Dirichlet, Neumann and Robin boundary conditions are allowed and the results hold for weak solutions in relevant subspaces of Hilbert - Sobolev space asso- ciated with the problem. The solutions of these problems are shown to be strong limits of finite rank perturbations of the fundametal solution of the problem. For exterior regions these expressions generalize multipole expansions. 1. Introduction This paper will describe some different representations of the Green’s functions (or solution operators) for Laplacian boundary value problems. The representations hold when Dirichlet, Neumann or Robin conditions are imposed and for exterior regions as well as on bounded domains with Lipschitz boundaries. The representations involve the fundamental solution of the Laplacian and the Steklov eigenfunctions of the Laplacian and are shown to converge in various Sobolev-type norms and follow from the construction of orthogonal bases of relevant Sobolev-Hilbert spaces. The results may be related to the methods described in the classic text of Bergman and Schiffer [10]. We concentrate our attention on how the Green’s functions differ from the fundamental solution of the differential operator. That is we seek to describe the boundary correction (BC) kernel B(., .) for various operators and boundary conditions such that G(x, y) = Γ(x, y) − B(x, y) for (x, y) ∈ Ω × Ω. (1.1) Here G, Γ are the Green’s function and the fundamental solution respectively. The (integral) operator B associated with this BC kernel is shown to be the limit of finite rank kernels involving the Steklov eigenfunctions and their single and double layer potentials. These approximations converge in H 1 −norms and, in general, are not L 2 − orthogonal expansions. It is of particular interest to note that this analysis applies to boundary value problems in exterior regions where the standard Green’s function may not be represented using eigenfunctions. Date : July 6, 2016. The author gratefully acknowledges research support by NSF award DMS 11008754. 2010 Mathematics Subject classification. Primary 35J08, Secondary 31B05, 35J05, 35P15. Key words and phrases. Greens Functions, Laplacian boundary value problems, Steklov eigenfunctions, multipole expansions, layer potentials. 1