Representations of Solutions of Laplacian Boundary

Authors: Giles Auchmuty Qi Han

Publish Date: 2013/09/12

Volume: 69, Issue: 1, Pages: 21-45

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Abstract

This paper treats the wellposedness and representation of solutions of Poisson’s equation on exterior regions UsubsetneqmathbbRN with N≥3 Solutions are sought in a space E 1U of finite energy functions that decay at infinity This space contains H 1U and existenceuniqueness theorems are proved for the Dirichlet Robin and Neumann problems using variational methods with natural conditions on the data A decomposition result is used to reduce the problem to the evaluation of a standard potential and the solution of a harmonic boundary value problem The exterior Steklov eigenproblems for the Laplacian on U are described The exterior Steklov eigenfunctions are proved to generate an orthogonal basis for the subspace of harmonic functions and also of certain boundary trace spaces Representations of solutions of the harmonic boundary value problem in terms of these bases are found and estimates for the solutions are derived When U is the region exterior to a 3d ball these Steklov representations reduce to the classical multipole expansions familiar in physics and engineering analysisLet D1 mathbbRN be the space of all realvalued Borelmeasurable functions on mathbbRN that decay at infinity are in L1 loc mathbbRN and have squareintegrable weak gradients This space is studied in Sects 32 and 82 of Lieb and Loss 17

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