I know how to define $W_0^{1,p}(\Omega)$, $\Omega\subset R^N$ open bounded smooth boundary, for any $1\leq p<\infty$. However, for definition of $W_0^{1,\infty}(\Omega)$, I always confused. Recently I am studying a research topic consult about Quasi-convex function, i.e., the function $f$: $R^{d\times N}\to R$ satisfies $$f(\xi)\leq \int_\Omega f(\xi+\nabla \phi(x))\,dx $$ for all $\phi(x)\in W_0^{1,\infty}(\Omega; R^d)$.
For example, we could find this definition in Dacorogna's book. However, Dacorogna never specific the definition of $W_0^{1,\infty}(\Omega;R^d)$. But I was keep wondering that what should be the proper definition of $W_0^{1,\infty}(\Omega;R^d)$. Should we just take $C_c^{\infty}(\Omega;R^d)$ and consider the closure in $W^{1,\infty}$ norm? I don't think so because this will make $W_0^{1,\infty}$ necessary to be in $C^1(\Omega)$, too strong.
I then find an definition in Leoni's book it states that we may use the $W^{1,\infty}$ weak-star closure instead of $W^{1,\infty}$ norm closure. But I failed the see what function do I have if I use $W^{1,\infty}$ weak-star closure. Do I still have $T[u]=0$? where $T$ is the trace operator.
