title: Weakly harmonic function
text: # 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 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.
  
  