In mathematics, a function f {\displaystyle f} is weakly harmonic in a domain D {\displaystyle D} if
∫ D f Δ g = 0 {\displaystyle \int _{D}f\,\Delta g=0}
for all g {\displaystyle g} with compact support in D {\displaystyle 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.
See also Weak solution Weyl's lemma
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