Weakly harmonic function

In mathematics, a function $$f$$ is weakly harmonic in a domain $$D$$ if
 * $$\int_D f\, \Delta g = 0$$

for all $$g$$ with compact support in $$D$$ and continuous second derivatives, where &Delta; 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.