Let $\Omega\subset\mathbb{R}^N$ be a bounded convex domain. Once every convex function is locally Lipschitz, we have that $\partial\Omega$ is Lipschitz, therefore, the trace operator $T: W^{1,2}(\Omega) \rightarrow L^2(\partial\Omega)$ is well defined.
Suppose that $\Omega \subset \{ x\in\mathbb{R}^N:\ x= (x_1,...,x_n),\ x_1 >0\}$ and $K \subset \partial \Omega$ is a compact set, with smooth boundary such that $K\subset \{ x\in\mathbb{R}^N : =(0,x_1,...,x_n)\}$.
Define $h(x) = 1$ if $x \in K$ and $h(x)=0$ if $ x \in \partial \Omega \setminus K$.
Is there a function $w \in W^{1,2}(\Omega)$ with $T(w) = h$? Intuitively this is true, but I don't know how to prove. Someone could help me to prove or disprove it? Thanks in advance!