Weakly harmonic function

{{refimprove|date=April 2023}}

In mathematics, a function $f$ is weakly harmonic in a domain $D$ if

:$\backslash int\_D\; f\backslash ,\; \backslash Delta\; g\; =\; 0$

for all $g$ with compact support in $D$ and continuous second derivatives, where Δ is the Laplacian.{{cite book |last1=Gilbarg |first1=David |last2=Trudinger |first2=Neil S. |title=Elliptic partial differential equations of second order |date=12 January 2001 |publisher=Springer Berlin Heidelberg |isbn=9783540411604 |page=29 |url=https://books.google.com/books?id=eoiGTf4cmhwC |access-date=26 April 2023}} 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.