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$u$ is a function in $W^{1,p}(\Omega)$ for $1\le p \le +\infty$, and its first derivative $\partial u/\partial x^i$ is in $C(\Omega)$( the space of continuous functions on open set(a domain) $\Omega$), then how to show $u$ is in $C^1(\Omega)$?

stephenkk
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  • Can you find an antiderivative and show that it equals $u$ (except possibly on a set of measure zero)? – RideTheWavelet May 21 '18 at 05:01
  • My idea is the same as yours. But the point is how to construct this something anti-derivative. – stephenkk May 21 '18 at 08:05
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    This has been answered here before. Take a look here, for example. – Glitch May 21 '18 at 13:48
  • Note that in the link @Glitch cited (+1, by the way), the function is assumed continuous. Here, the function $u$ is only assumed to be in $W^{1,p},$ hence only defined up to sets of measure zero, and therefore we can only say that $u$ is equal a.e. to an element of $C^{1}(\Omega)$. – RideTheWavelet May 21 '18 at 15:41

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