Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class

Solution 1:

"Locally" is ambiguous here

$f$ is locally Lipschitz in $\Omega$ if and only if $f \in W^{1,\infty}_{loc}(\Omega)$

The validity of this claim depends on interpretation of "locally Lipschitz". Does it mean

1. every point of $\Omega$ has a neighborhood in which $f$ is Lipschitz, or
2. there is $L$ such that every point of $\Omega$ has a neighborhood in which $f$ is $L$-Lipschitz (i.e., satisfies $|f(x)-f(y)|\le L|x-y|$).

With interpretation 1) the above would be false, because $f(x)=1/x$ is locally Lipschitz on the interval $(0,1)$. The authors meant interpretation 2, but you should be aware that it may be less common. Saying "locally $L$-Lipschitz" would be more precise.

Counterexample and quasiconvexity

$W^{1,\infty}(\Omega) = C^{0,1}(\Omega)$

This is not always true. For example, let $\Omega$ be the plane with the slit along negative $x$-axis. Using polar coordinates $r,\theta$, with $-\pi<\theta<\pi$, define $u(r,\theta) = r\theta$. You can check that this is a $W^{1,\infty}$ function (it is locally $10$-Lipschitz, say), but it is not in $C^{0,1}(\Omega)$ because the values of $f$ just above the slit and just below it are far apart.

The counterexample is taken from here, where you can also find a result in the positive direction; for sufficiently nice $\Omega$ the equality holds. Also, see the answer and references in relation between $W^{1,\infty}$ and $C^{0,1}$.

Vanishing on the boundary

The definition of $W^{1,\infty}_0(\Omega)$ is a bit tricky since the usual approach (complete the space of smooth compactly supported functions with respect to the Sobolev norm) does not apply. Instead one can define $W_0^{1,\infty}(\Omega)$ as follows. Every element of $W^{1,\infty}(\Omega)$ has a continuous representative $u$. If $u$ satisfies $\lim_{x\to a}u(x)= 0$ for every $a\in \partial \Omega$, then we say that $u\in W_0^{1,\infty}(\Omega)$. This is a closed subspace, because convergence in $W^{1,\infty}$ norm implies uniform convergence.

As an aside: one can give a unified definition of $W^{1,p}_0(\Omega)$ that works for all $1\le p\le \infty$: a function is in $W^{1,p}_0(\Omega)$ if its zero extension to $\mathbb R^n$ is in $W^{1,p}(\mathbb R^n)$.

For every domain, $W_0^{1,\infty}(\Omega)$ is the same as the set of Lipschitz functions on $\Omega$ that tend to $0$ at the boundary. Indeed, we can extend by zero to the rest of $\mathbb R^n$ and use the fact that $\mathbb R^n$ is convex.