Let $\Omega$ be a domain of $R^n,$ let $\omega$ be open subset of $\Omega$ and let $\theta \in W^{2,\infty}(\omega).$
I am wondering about the existence of a function $\tilde{\theta} \in W^{2,\infty}(\Omega)$ (eventually, under some conditions on the value of $\theta$ on $\partial \omega) $ such that :
1) $\tilde{\theta}=\theta $ on $\omega,$
2) $\Delta \tilde{\theta}=0$ on $\Omega-\omega.$
Thanks!