Weakly harmonic function
≈ 2 min read
Names in other languages
- Portuguêsfunção fracamente harmónica
- Portuguêsfunção fracamente harmônica
- eomalforte harmonia funkcio
- Englishweakly harmonic function
- 日本語弱調和函数
Article bodies are English and are not machine-translated. A name in another language is not a translation of the article.
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. 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.
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. 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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