Let $\,D\subseteq\mathbb{R}^{n-1}$ be a convex bounded domain. Let$A:D\to(0,\infty)$ be a Lipschitz continuous function. Let $\,\Omega\,$ be bounded domain in $\,\mathbb{R}^{n}\,$ of the form \begin{equation} \Omega:=\left\{(x',x_{n})\in\mathbb{R}^{n}|\,x'\in D,\,0\leqslant x_{n}<A(x')\right\}. \end{equation}

Let $u\in C^{1}(\overline{\Omega})$ be such that \begin{equation} u=0\quad\mbox{on}\quad \overline{\Omega} \cap\{x_{n}=0\}. \end{equation} QUESTION: Let $p>1$ and let $\,\mathbb{R}_{+}^{n}=\{x\in\mathbb{R}^{n}|\,x_{n}>0\}.\,$ Given $\,\epsilon>0,\,$ does there exist a function $\,\tilde{u}\in W_{0}^{1,p}(\mathbb{R}_{+}^{n})\,$such that \begin{equation} \tilde{u}|_{\Omega}=u, \mbox{and}\quad \int_{\mathbb{R}_{+}^{n}\setminus\Omega}|\nabla\tilde{u}|^{p}\,dx<\epsilon. \end{equation}