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The Helmholtz–Weyl decomposition in weighted Sobolev spaces
Author(s) -
Zaja̧czkowski Wojciech M.
Publication year - 2010
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.1347
Subject(s) - mathematics , sobolev space , decomposition , divergence (linguistics) , pure mathematics , helmholtz free energy , cartesian coordinate system , mathematical analysis , geometry , physics , quantum mechanics , ecology , linguistics , philosophy , biology
Let f ∈ L 2, − µ (ℝ 3 ), wherewhere x = ( x 1 , x 2 , x 3 ) is the Cartesian system in ℝ 3 , x ′ = ( x 1 , x 2 ), , µ∈ℝ + \ℤ. We prove the decomposition f = − ∇ u + g , with g divergence free and u is a solution to the problem in ℝ 3Given f ∈ L 2, − µ (ℝ 3 ) we show the existence of u ∈ H 1 −µ (ℝ 3 ) such thatwhereSince f, u, g are defined in ℝ 3 we need a sufficiently fast decay of these functions as | x |→∞. Copyright © 2010 John Wiley & Sons, Ltd.