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I am trying to prove the Sobolev extension theorem for $W^{1,\infty}$. However, I am currently stuck and searching for a book that contains a proof of this theorem. Unfortunately, I haven't been able to find any references so far.

Currently, I am attempting to understand the proof provided at Extension Theorem for the Sobolev Space $W^{1, \infty}(U)$. However, I have some questions regarding certain steps in the proof.

Firstly, I don't understand why $\int_{B^-}\overline{u}\varphi_{x_n}dx=\int_{B^+}u(y',y_n)\varphi_{x_n}(y',-y_n)dy$, and why $\int_{B^+}-u_{y_n}(y',y_n)\varphi(y',-y_n)dy=\int_{B^-}-u_{x_n}(x',-x_n)\varphi(x',x_n)dy$.

Additionally, when using integration by parts, I obtained $\int_{B^+}u\varphi_{x_n}dx=\int_{x_n}u\varphi d S_x-\int_{B^+}u_{x_n}\varphi dx$. However, in the proof, it seems to suggest that $\int_{B^+}u\varphi_{x_n}dx=-\int_{x_n}u\varphi d S_x-\int_{B^+}u_{x_n}\varphi dx$. I'm confused about this discrepancy.

If anyone has a reference with a complete and clear proof of the Sobolev extension theorem, I would greatly appreciate it.

scottish
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  • If you just want a proof of such an extension, you could argue as follows: If $U$ has a $C^1$ boundary, then $W^{1,\infty}(U)$ is exactly (up to representatives of course) the class of bounded Lipschitz functions and extending Lipschitz functions is "easy" (see for example this question for references). I believe all these results are in Evans' PDE book. – Jose27 Sep 04 '23 at 16:58

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