Weakly harmonic function
Template:Refimprove In mathematics, a function <math>f</math> is weakly harmonic in a domain <math>D</math> if
- <math>\int_D f\, \Delta g = 0</math>
for all <math>g</math> with compact support in <math>D</math> and continuous second derivatives, where Δ is the Laplacian.<ref name="gilbarg">Template:Cite book</ref> 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.