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!

  • 1
    $\begingroup$ You should be interested by Rademacher's theorem. $\endgroup$ – Davide Giraudo Oct 4 '12 at 10:23
  • $\begingroup$ Indeed I should. Thank you! $\endgroup$ – Sh4pe Oct 4 '12 at 10:29

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

  • $\begingroup$ I think OP might have been more interested in Theorem 3 from page 277, in particular statement (ii), since he asks for $L^p$ rather than $L^\infty$ and the arguments are rather different. $\endgroup$ – F.Webber Jan 21 '17 at 22:23
  • $\begingroup$ I might be looking at a different version of Evans but for me the relevant page was p. 294. In any case, people who don't have access to Evans might be interested in math.stackexchange.com/questions/169630/… where a few more details were spelled out. $\endgroup$ – balu Nov 11 '19 at 2:55

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