Weakly harmonic function

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

$\underset{D}{\int }f\mathrm{\Delta }g=0$

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.

See also

• Weak solution
• Weyl's lemma

References

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