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Well‐posedness and approximation of solutions of linear divergence‐form elliptic problems on exterior regions
Author(s) -
Auchmuty Giles,
Han Qi
Publication year - 2015
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3199
Subject(s) - mathematics , boundary value problem , divergence (linguistics) , dirichlet distribution , mathematical analysis , eigenfunction , space (punctuation) , harmonic function , neumann boundary condition , pure mathematics , eigenvalues and eigenvectors , philosophy , linguistics , physics , quantum mechanics
This paper describes well‐posedness, spectral representations, and approximations of solutions of uniformly elliptic, second‐order, divergence form elliptic boundary value problems on exterior regions U inR Nwhen N ≥ 3. Inhomogeneous Dirichlet, Neumann, and Robin boundary conditions are treated. These problems are first shown to be well‐posed in the space E 1 ( U ) of finite‐energy functions on U using variational methods. Spectral representations of these solutions involving Steklov eigenfunctions and solutions subject to zero Dirichlet boundary conditions are described. Some approximation results for the A ‐harmonic components are obtained. Positivity and comparison results for these solutions are given. Copyright © 2014 John Wiley & Sons, Ltd.