weak derivative
Let f:Ω→𝐑 and g=(g1,…,gn):Ω→𝐑n be locally integrable functions defined on an open set Ω⊂𝐑n. We say that g is the weak derivative of f if the equality
∫Ωf∂ϕ∂xi=-∫Ωgiϕ |
holds true for all functions ϕ∈𝒞∞c(Ω) (smooth functions with compact support in Ω) and for all i=1,…,n. Notice that the integrals involved are well defined since ϕ is bounded with compact support and because f is assumed to be integrable on compact subsets of Ω.
Comments
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1.
If f is of class 𝒞1 then the gradient of f is the weak derivative of f in view of Gauss Green Theorem. So the weak derivative is an extension
of the classical derivative
.
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2.
The weak derivative is unique (as an element of the Lebesgue space L1loc) in view of a result about locally integrable functions.
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3.
The same definition can be given for functions with complex values.
Title | weak derivative |
---|---|
Canonical name | WeakDerivative |
Date of creation | 2013-03-22 14:54:52 |
Last modified on | 2013-03-22 14:54:52 |
Owner | paolini (1187) |
Last modified by | paolini (1187) |
Numerical id | 16 |
Author | paolini (1187) |
Entry type | Definition |
Classification | msc 46E35 |
Related topic | SobolevSpaces |