Weakly harmonic function explained

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.^[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

Notes and References

1. Book: Gilbarg . David . Trudinger . Neil S. . Elliptic partial differential equations of second order . 12 January 2001 . Springer Berlin Heidelberg . 9783540411604 . 29 . 26 April 2023.