Weakly harmonic function

In mathematics, a function [math]\displaystyle{ f }[/math] is weakly harmonic in a domain [math]\displaystyle{ D }[/math] if

[math]\displaystyle{ \int_D f\, \Delta g = 0 }[/math]

for all [math]\displaystyle{ g }[/math] with compact support in [math]\displaystyle{ D }[/math] and continuous second derivatives, where Δ is the Laplacian. This is the same notion as a weak derivative, however, a function can have a weak derivative and not be differentiable. In this case, we have the somewhat surprising result that a function is weakly harmonic if and only if it is harmonic. Thus weakly harmonic is actually equivalent to the seemingly stronger harmonic condition.

See also

• Weak solution
• Weyl's lemma