Weakly harmonic function

In mathematics, a function is weakly harmonic in a domain if

for all with compact support in 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

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References

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  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.