Extension of Sobolev function

Question

Let $D$ be a convex bounded domain in $\mathbb{R}^{n-1}$. Let $A:D\to\mathbb{R}^{+}$ be a Lipschitz continuous function. Let $\,\Omega\,$ be a bounded domain in $\mathbb{R}^{n}$ of the form $\,\Omega=\left\{(x',x_{n})\in\mathbb{R}^{n}\,|\,\,x'\in D,\,0<x_{n}<A(x')\right\}\,$. Let $u\in C^{1}(\overline{\Omega})$ and $\,u=0\,$ on $\,\overline{\Omega}\cap\{x_{n}=0\}\,$. Question: Given any $\epsilon>0,$ does there exist a function $\,\tilde{u}\,$(extension of $u$) such that $\tilde{u}|_{\Omega}=u$, $\,\tilde{u}\in W_{0}^{1,p}(\mathbb{R}_{+}^{n})\,$ for $p>1$ and $\int_{R_{+}^{n}\setminus\Omega}|\nabla\tilde{u}|^{p}\,dx<\epsilon\,$?