The G-N-S inequality can be stated as follows:
Let $U\subset\mathbb R^d$, open bounded, with $C^1$ boundary, then for any $w\in W^{k,p}_0(U)$, $p<d$ $$\|w\|_{L^{p^*}(U)}\le C(d)\|\nabla w\|_{L^2(U)}$$ (Where $p^*=\frac{dp}{d-p}$).
But can in be used for $u\in W^{k,p}(V)$, where $V\subset\subset U$, (compactly contained)? By taking $W$ such that $V\subset\subset W\subset\subset U$, and extending $u$ by $0$ to $\tilde{u}$ in $W$, and considering, $\tilde{u}\in W^{k,p}_0(W)$, and applying G-S-N inequality to $\tilde{u}$, i.e.
$\|u\|_{L^{p^*}(V)}=\|\tilde{u}\|_{L^{p^*}(W)}\le C(d)\|\nabla\tilde u\|_{L^p(W)}=C(d)\|\nabla u\|_{L^p(V)}$
Is this idea correct?
Thanks in advance for any help.
Or is the best that is possible is to use the general sobolev inequality on $u\in W^{1,p}(V)$? where by for $q$ such that $\frac{1}{q}=\frac{1}{p}-\frac{1}{d}$, i.e. $q=p^*$, we have $$\|u\|_{L^q(V)}=\|u\|_{L^{p^*}(V)}\le C\|u\|_{W^{1,p}(V)}$$