On Helmholtz’s decomposition theorem and Poisson’s equation with an infinite domain
Ton Tran Cong · Quarterly of Applied Mathematics · 1993
strong version).The theorem can be proved using the identity V w = V(V • w) -V x (V x w) where w(x) satisfies V w = u.However, such a simple proof, as given in vector analysis (see, e.g., Lass [4, p. 156], Aris [5, p. 70], or Bowen and Wang [6, 2 s p. 328]), requires u(x) or V • u to be of order 0, at infinity when the region DcJi3 under consideration is infinite.Various authors have attempted to avoid or relax such restriction.Phillips [7, p. 186], and Weatherburn [8, p. 74], used a more complicated application of the solution to Poisson's equation to relax i c the restriction to |u(x)| = 0(|x| ), S > 0, at infinity.Blumenthal [9] devised a method of accelerating the convergence of the solution to Poisson's equation and proved that every function u(x) e C°°(D) bounded at infinity by 0(log|x|) can be decomposed into a curl-free vector and a divergence-free vector, which are also in C°°(D).Gurtin [10] applied the method to prove that every u(x) e C°(D U dD) n Cl(D -dD) bounded at infinity by (9(|x|~ 0, can be written as V8 + V x b for some 0(x), b(x) e c'(D -dD).We also have another line of approach to this decomposition problem (Nikodym [11], Friedrichs [12], Weyl [13], Bykhovski and Smirnov [14], and Fujiwara and Morimoto [15]), complementing the classical