structural:summarization, information:highlighting
Weak formulation

The Helmholtz decomposition can be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). Suppose Ω is a bounded, simply-connected, Lipschitz domain. Every square-integrable vector field u ∈ (L (Ω)) has an orthogonal decomposition:

u = ∇ φ + ∇ × A {\displaystyle \mathbf {u} =\nabla \varphi +\nabla \times \mathbf {A} }

where φ is in the Sobolev space H (Ω) of square-integrable functions on Ω whose partial derivatives defined in the distribution sense are square integrable, and 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 u ∈ H (curl, Ω), a similar decomposition holds:

u = ∇ φ + v {\displaystyle \mathbf {u} =\nabla \varphi +\mathbf {v} }

where φ ∈ H (Ω), v ∈ (H (Ω)).