Classical Derivative, Weak Derivative and Integration by Parts - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:02:07Z http://mathoverflow.net/feeds/question/116580 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116580/classical-derivative-weak-derivative-and-integration-by-parts Classical Derivative, Weak Derivative and Integration by Parts tatin 2012-12-17T09:59:42Z 2012-12-17T21:18:55Z <p>Hello,</p> <p>While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated. </p> <p><strong>QUESTION</strong></p> <p>Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function $f\in L^1_{\text{loc}}(U)$ such that </p> <p>1) the classical derivative $Df$ exists everywhere in $U$.</p> <p>2) $f$ is weakly differentiable in $U$. Let us write $D_w f$ to denote the weak derivative of $f$.</p> <p>3) $Df\neq D_w f$, on a set of <strong>positive measure</strong>.</p> <p><strong>Note that, we are assuming the existence of both the derivatives.</strong> I'm aware of examples where one exists while other one does not. </p> <p>The problem seems to be related to the question of validity of integration by parts for functions that are only differentiable. </p> <p>Thank you.</p> http://mathoverflow.net/questions/116580/classical-derivative-weak-derivative-and-integration-by-parts/116642#116642 Answer by Daniel Spector for Classical Derivative, Weak Derivative and Integration by Parts Daniel Spector 2012-12-17T20:17:22Z 2012-12-17T21:18:55Z <p>Suppose $f \in W^{1,1}_{loc}(U)$. Then no, since for such an $f$, we have that $Df$ exists and the approximate limit</p> <p>$ap\lim_{y\to x} \frac{f(x)-f(y)-Df(x)(x-y)}{|x-y|} = 0$</p> <p>exists for almost every $x$, while from assuming classical differentiability we have</p> <p>$\lim_{y\to x} \frac{f(x)-f(y)-\nabla f(x)(x-y)}{|x-y|} = 0$</p> <p>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.</p> <p><a href="http://www.encyclopediaofmath.org/index.php/Approximate_differentiability" rel="nofollow">http://www.encyclopediaofmath.org/index.php/Approximate_differentiability</a></p>