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I am struggling with Evans PDE problem 5.10. #12. Here is the question: Let $V\subset \subset U$ be open sets. Show by example that if we have $\|D^hu\|_{L^1(V)}\le C$ for all $0< |h|<\frac{1}{2}\operatorname{dist}(V,\partial U)$, it does not necessarily follow that $u\in W^{1,1}(V)$.

I have no idea why the bigger open set $U$ is relevant. I would very much appreciate your help.

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    $U$ is there just to make sure the difference quotient in $V$ is well-defined.2012-11-11
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    I see, but am still in trouble proving this.2012-11-12
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    Maybe use the following idea. The assumption gives a conditions for elements why are not too near of the boundary of $U$, but doesn't prevent a bad behaviour near it.2012-11-12

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