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Let $\Omega\in\mathbb{R}^n$ be open and bounded. I'm wondering if a function $f\in C^{0,1}(\Omega)$ (a Lipschitz-continuous one) is also an element of $W^{1,2}(\Omega)$ (that is the space of weakly derivatives functions whose first weak derivatives are $L^2$-functions).

One can easily show that $\|f\|_{L^2}$ is bounded. What I did not yet manage to show is that the weak derivatives $\partial_{x_i}f$ exist for $i=1,\dots,n$.

Do they even exist? And if so, is there a constant $C$ such that $\|f\|_{C^{0,1}} \le C\|f\|_{W^{1,2}}$ or $\|f\|_{W^{1,2}}\le C\|f\|_{C^{0,1}}$.

I'd be glad for any help or hints to literature on this.

Thank you very much!

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You should be interested by Rademacher's theorem. – Davide Giraudo Oct 4 '12 at 10:23
Indeed I should. Thank you! – Sh4pe Oct 4 '12 at 10:29
up vote 2 down vote accepted

Take a look in the page 279 of this book: "Evans - Partial Differential Equation".

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Great! Thank you! – Sh4pe Oct 4 '12 at 13:43
You're welcome. – Tomás Oct 4 '12 at 13:54

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