Classical Derivative, Weak Derivative and Integration by Parts

2012-12-17T09:59:42Z

Hello,

While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated.

QUESTION

Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function $f\in L^1_{\text{loc}}(U)$ such that

1) the classical derivative $Df$ exists everywhere in $U$.

2) $f$ is weakly differentiable in $U$. Let us write $D_w f$ to denote the weak derivative of $f$.

3) $Df\neq D_w f$, on a set of positive measure.

Note that, we are assuming the existence of both the derivatives. I'm aware of examples where one exists while other one does not.

The problem seems to be related to the question of validity of integration by parts for functions that are only differentiable.

Thank you.

Answer by Daniel Spector for Classical Derivative, Weak Derivative and Integration by Parts

2012-12-17T20:17:22Z

Suppose $f \in W^{1,1}_{loc}(U)$. Then no, since for such an $f$, we have that $Df$ exists and the approximate limit

$ap\lim_{y\to x} \frac{f(x)-f(y)-Df(x)(x-y)}{|x-y|} = 0$

exists for almost every $x$, while from assuming classical differentiability we have

$\lim_{y\to x} \frac{f(x)-f(y)-\nabla f(x)(x-y)}{|x-y|} = 0$

exists for every $x \in U$. In particular, the classical differential is a candidate for the approximate differential, and so $Df=\nabla f$ wherever the two exist, and hence in $U$ up to a set of measure zero.

http://www.encyclopediaofmath.org/index.php/Approximate_differentiability

Classical Derivative, Weak Derivative and Integration by Parts - MathOverflow

Classical Derivative, Weak Derivative and Integration by Parts

Answer by Daniel Spector for Classical Derivative, Weak Derivative and Integration by Parts