Why if $u_h \rightharpoonup u$ in $*w-W^{1,\infty}$ then exists a subsequence s.t. $u_{h_k}\rightarrow u$ in $L^{\infty}$?

Question

I can't understand why the following fact holds:

I consider a sequence $(u_h)\subset W^{1, \infty}(U, \mathbb{R}^N)$, with $U$ open bounded set in $\mathbb{R}^n$, such that $$u_h \rightharpoonup u$$ $*w-W^{1,\infty}$ (the * weak convergence).

Then it is possible to extract a subsequence such that $u_{h_k}\rightarrow u$ in $L^{\infty}$

Edit

Thanks to @daw 's answer I know how to prove this fact under the hypothesis $U$ bounded and with Lipschitz boundary.

I would like to know if this fact is true also if I don't assume that the boundary of $U$ is Lipschitz.

Thanks for the help!

Answer 1

This argument works if $U$ is bounded and has Lipschitz boundary.

Since $(u_h)$ converges weak-star in $W^{1,\infty}(U)$, it is bounded in $W^{1,\infty}(U)$ and also bounded in $W^{1,n+1}(U)$ since $U$ is bounded. The space $W^{1,n+1}(U)$ is continuously embeddded into the space of H"older continuous functions $C^{0,1/n}(\bar U)$, Morrey embedding theorem. This space compactly embeds into $C(\bar U)$, thus also into $L^\infty(\Omega)$ by Arzela-Ascoli. So we have this chain of continuous $\hookrightarrow$ and compact $\hookrightarrow\hookrightarrow$ embeddings: $$ W^{1,\infty}(U) \hookrightarrow W^{1,n+1}(U) \hookrightarrow C^{0,1/n}(\bar U)\hookrightarrow\hookrightarrow C(\bar U) \hookrightarrow L^\infty(U). $$

Answer 2

To answer your additional question: Without assumptions on the boundary the statement is false. You can pretty much do the same thing as in the unbounded case. More precisely: look at the bounded part of a spiral (with some thickness) with infinite length. Parameterized along the length of the spiral from the outside in, just let $f_n=0$ on $[0,n]$ and $f_n = sin(\pi x)$ for $x>n$. This converges weak-$\ast$ to $0$ in $W^{1,\infty}$, but clearly not (essentially) uniformly. (The boundary is not Lipschitz in the center of the spiral.)