weak derivative

Let f:Ω𝐑 and g=(g1,,gn):Ω𝐑n be locally integrable functions defined on an open set Ω𝐑n. We say that g is the weak derivative of f if the equality


holds true for all functions ϕ𝒞c(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.


  1. 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 extensionPlanetmathPlanetmath of the classical derivativePlanetmathPlanetmath.

  2. 2.

    The weak derivative is unique (as an element of the Lebesgue space Lloc1) in view of a result about locally integrable functions.

  3. 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