Weakly harmonic function

In mathematics, a function ${\displaystyle f}$ is weakly harmonic in a domain ${\displaystyle D}$ if

${\displaystyle \int _{D}f\,\Delta g=0}$

for all ${\displaystyle g}$ with compact support in ${\displaystyle D}$ and continuous second derivatives, where Δ is the Laplacian.^[1] 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

References

1. Gilbarg, David; Trudinger, Neil S. (12 January 2001). Elliptic partial differential equations of second order. Springer Berlin Heidelberg. p. 29. ISBN 9783540411604. Retrieved 26 April 2023.