Editing Helmholtz decomposition (section)

=== Weak formulation ===
The Helmholtz decomposition can be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). Suppose {{math|Ω}} is a bounded, simply-connected, [[Lipschitz domain]]. Every [[square-integrable]] vector field {{math|'''u''' ∈ (''L''<sup>2</sup>(Ω))<sup>3</sup>}} has an [[orthogonality|orthogonal]] decomposition:<ref name="amrouche1998" /><ref name="dautray1990" /><ref name="girault1986" />

<math display="block">\mathbf{u}=\nabla\varphi+\nabla \times \mathbf{A}</math>

where {{mvar|φ}} is in the [[Sobolev space]] {{math|''H''<sup>1</sup>(Ω)}} of square-integrable functions on {{math|Ω}} whose partial derivatives defined in the [[distribution (mathematics)|distribution]] sense are square integrable, and {{math|'''A''' ∈ ''H''(curl, Ω)}}, the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl.

For a slightly smoother vector field {{math|'''u''' ∈ ''H''(curl, Ω)}}, a similar decomposition holds:

<math display="block">\mathbf{u}=\nabla\varphi+\mathbf{v}</math>

where {{math|''φ'' ∈ ''H''<sup>1</sup>(Ω), '''v''' ∈ (''H''<sup>1</sup>(Ω))<sup>''d''</sup>}}.