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In Partial Differential Equation by Lawerence Evan p284 there is this theorem stated:

Let $U$ be a bounded open subset of $\mathbb{R}^n$ with $C^1$ boundary. Suppose $u\in W^{k,p}$ then if $k>n/p$ we have

$ u\in C^{\alpha, \gamma}(\overline{U}) $ where $\alpha = k-\left[n/p\right]-1$ and $\gamma = \left[n/p\right]+1-n/p$ if $n/p$ is not an integer and any $0<\gamma<1$ if $n/p\in\mathbb{N}$.

I have two questions:

  1. Does this result extend to $U$ being an open subset with only Lipschitz boundary?

2.Does the result also holds $k\not\in\mathbb{N}$? The author doesn't mention anyway that $k$ should be an integer but I just wanted to check.

Thank you in advance.

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1 Answer 1

up vote 1 down vote accepted

1- The Sobolev embedding is proved first in the case $U=\mathbb R^n$, and then for a general $U$ by using a right inverse $j$ of the restriction operator $W^{s,p}(\mathbb R^n)\rightarrow W^{k,p}(U)$. When $U$ is a half-space, a convenient $j$ is Babitch's extension. When the boundary is smooth, use an atlas to reduce to the previous case.

The situation is however not easy if the boundary is only Lipschitz. If $U$ is a (hyper-)rectangle, then $j$ can be build by applying $n-1$ times the Babitch operator. Thus the Sobolev embedding holds true in this case, whatever $\alpha$ is. Still true if the boundary is smooth, up to some kinks with angle $\frac{\pi}{m}$ with $m$ an integer. It is less clear for more general domains with Lipschitz boundary.

2- Sobolev embeddings hold even if $k$ is not an integer. The definition of the Sobolev space is then a bit more involved. What really matters is that $\beta:=k-\frac{n}{p}$ is not an integer. Then $\alpha=[\beta]$ and $\gamma=\beta-\alpha$. And of course regularity of the boundary.

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