In mathematics, a function
f
is weakly harmonic in a domain
D
if
∫
D
f
Δ
g
=
0
for all
g
with compact support in
D
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.