Abstract
This paper describes different representations for solution operators of Laplacian boundary value problems on bounded regions in \({\mathbb R}^N, N \ge 2\) 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 associated with the problem. The solutions of these problems are shown to be strong limits of finite rank perturbations of the fundamental solution of the problem. For exterior regions these expressions generalize multipole expansions.
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Acknowledgments
The author gratefully acknowledges research support by NSF Award DMS 11008754.
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Auchmuty, G. Steklov Representations of Green’s Functions for Laplacian Boundary Value Problems. Appl Math Optim 77, 173–195 (2018). https://doi.org/10.1007/s00245-016-9370-4
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DOI: https://doi.org/10.1007/s00245-016-9370-4
Keywords
- Greens functions
- Laplacian boundary value problems
- Steklov eigenfunctions
- Multipole expansions
- Layer potentials