An Introduction to Sobolev Spaces

Weak Derivative

Author(s): Erhan Pişkin* and Baver Okutmuştur * .

Pp: 86-96 (11)

DOI: 10.2174/9781681089133121010004

* (Excluding Mailing and Handling)


This chapter is dedicated to applications of the weak derivatives. We extend the classical knowledge of derivative and provide basic properties of weak derivatives in this part. The concept of the weak derivative can be considered as a generalization of the classical derivative. More precisely there are functions which are not differentiable in the classical sense but weakly differentiable. Further properties and analysis are given in the examples of this section. For the applications of weak derivatives and further details, we refer the reader to [25–28].

Keywords: Weak derivative, classical derivative, β-th weak derivative, L1loc (Ω) space, multivariable derivative, generalized weak derivative, Heaviside function, Dirac-delta function.

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