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I was reading PDE by Evans and got confused about the proof of the general Sobolev inequalities. One part of the theorem states that if $U$ is a bounded open set in $\mathbb{R}^n$ with $C^1$ boundary, $k>n/p$ and $n/p$ is an integer, then $W^{k,p}(U)\subset C^{k-[n/p]-1,\gamma}(\bar{U})$ where $\gamma$ is any number $<1$.

The proof says by Morrey's inequality we have $D^\alpha u\in C^{0,1-n/r}$ for all $|\alpha|\leq k-l-1,$ so $u\in C^{k-l-1,1-n/r}.$ My question is that why the regularity can be added up directly, even though in the RHS, Holder spaces need the classical derivative, and LHS we only have weak derivative of $u$?

I understand that under this situation, for each first order weak derivative $\partial_i u$ of $u$, we have a version that is Holder continuous. And we can find a version of $u$ that has classical partial derivatives almost everywhere and it is equal to the corresponding weak derivative almost everywhere. But by these properties I can only see that $u$ is $C^1$ "almost everywhere". How do we get $u\in C^1$ entirely? Thanks for any help.

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Once $D^\alpha u \in C^{0,\gamma}$ the weak derivative is actually classical. The general Sobolev inequalities apply to a version of $u$, just like Morrey. So there is a version of $u$ that is in $C^{k,\gamma}$, but certainly other versions may be less regular.

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