Weak Derivatives and Their Role in the Construction of Sobolev Spaces

Abstract

This paper studies weak derivatives as a generalization of classical derivatives defined in an integral sense. By introducing test functions and  spaces.  We prove that weak derivatives coincide with classical derivatives for smooth functions and are unique up to sets of measure zero.  Through particular examples, we show that weak derivatives can exist even when classical differentiability fails, and we identify how discontinuity can prevent their existence.  We then define Sobolev spaces as functions with weak derivatives in  and prove that they are Banach spaces. Finally, we establish that weak derivatives keep the main properties of classical derivatives, such as linearity and Leibniz's formula, under mild assumption.