A Constructive Proof of Helmholtz's Theorem. Issue 4 (4th September 2019)

• Record Type: Journal Article
• Authors: De La Calle Ysern, Bernardo; Sabina De Lis, José C
• Is Part Of: Quarterly journal of mechanics and applied mathematics. Volume 72:Issue 4(2019:Nov.)
• Page Start: 521
• Page End: 533
• Publication Date: 2019-09-04
• Subjects: Mechanics -- Mathematics -- Periodicals; Applied mathematics -- Periodicals; 530.1
• DOI: 10.1093/qjmam/hbz016
• Languages: English
• ISSNs: 0033-5614

Abstract

Summary: It is a known result that any vector field ${\boldsymbol{u}}$ that is locally Hölder continuous on an arbitrary open set $\Omega\subset \mathbb{R}^3$ can be written on $\Omega$ as the sum of a gradient and a curl. Should $\Omega$ be unbounded, no conditions are required on the behaviour of ${\boldsymbol{u}}$ at infinity. We present a direct, self-contained proof of this theorem that only uses elementary techniques and has a constructive character. It consists in patching together local solutions given by the Newtonian potential that are then modified by harmonic approximations—based on solid spherical harmonics—to assure convergence near infinity for the resulting series.