Consider a BV function $u \in BV(\mathbb{R}^N; \mathbb{R}^N)$. What can be said about the difference quotient $$ \frac{u(x+\epsilon y)-u(x)}{\epsilon} $$ regarding its convergence as $\epsilon \to 0$? That is, in what sense does it converge to the derivative $Du = D^{a.c.} u + D^{sing} u$? Under what assumptions does the convergence hold almost everywhere?
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1$\begingroup$ What is the meaning of the quotient if N>1? $\endgroup$– Pietro MajerApr 13, 2019 at 20:18
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$\begingroup$ But then converging to anything as ||h|| goes to zero, seems problematic even for N=1 and u(x)=x $\endgroup$– Pietro MajerApr 13, 2019 at 21:31
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$\begingroup$ @PietroMajer Actually, I've edited the question now. $\endgroup$– RikuApr 13, 2019 at 23:58
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$\begingroup$ Now the quotient depends on $y$, but the putative limit does not …. $\endgroup$– LSpiceApr 14, 2019 at 0:58
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$\begingroup$ Maybe you want to consider $\frac{u(x+ty) - u(x)}{t}$ for fixed $y\in \mathbb R^N$ and to study it as a function of $x$ depending on a parameter $t$? $\endgroup$– SkeeveApr 14, 2019 at 6:42
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