Higher-order Rellich--Kondrachov compactness

If $\mathrm{\Omega }=\mathrm{\varnothing }$, all target spaces are trivial and the assertions hold. Otherwise choose the bounded extension $E$ at $(k,p)$ and a smooth cutoff $\eta$ equal to one near $\bar{\mathrm{\Omega }}$, with compact support in a ball $B$. By the weak Leibniz rule, $v_j=\eta E{u}_j$ is bounded in $W^{k,p}({\mathbb{R}}^n)$ and equals $u_j$ on $\mathrm{\Omega }$ (A Euclidean bump for a compact set inside an open set, Weak Leibniz rule with a smooth factor). For each $\mathrm{\mid }\alpha \mathrm{\mid }\le m$, $D^{\alpha }{v}_j$ is bounded in $W^{1,p}(B)$ by [F1]. The smooth ball is a $W^{1,p}$-extension domain (Bounded C^k domains admit integer-order Sobolev extension), so [F2] applied on $B$, followed by restriction to $\mathrm{\Omega }$, gives a common subsequence on which all $D^{\alpha }{u}_j$ converge in $L^p(\mathrm{\Omega })$. Only finitely many derivative sequences are extracted.

Consider cases (a) and (b). By [F3], for every $\mathrm{\mid }\alpha \mathrm{\mid }\le m$ the sequence $(D^{\alpha }{u}_{j_k})$ is bounded in $L^{p_{k-m}^{\ast }}(\mathrm{\Omega })$ in case (a), and in every finite $L^r(\mathrm{\Omega })$ in case (b). If $q\le p$, finite- measure inclusion [F4] and step 1.1 make each derivative sequence Cauchy in $L^q$. If $p<q<p_{k-m}^{\ast }$ in case (a), choose $r=p_{k-m}^{\ast }$; if $p<q<\mathrm{\infty }$ in case (b), choose any finite $r>q$. Lyapunov interpolation [F4] applied to differences, whose $L^p$ norms tend to zero by step 1.1 and whose $L^r$ norms are uniformly bounded by [F3], makes every derivative sequence Cauchy in $L^q$. Completeness of $L^q$ gives limits $v_{\alpha }$. Passing to the limit in the weak derivative test identities by [F5] shows that $v_{\alpha }=D^{\alpha }{v}_0$ for all $\mathrm{\mid }\alpha \mathrm{\mid }\le m$, so $u_{j_k}\to v_0$ in $W^{m,q}(\mathrm{\Omega })$. The higher-order embedding [F3] also gives boundedness of the inclusion into each stated target, so this is compact embedding.

Consider case (c), and fix $0\le \beta <\min \{1,k-m-\frac{n}{p}\}$. Choose ${\beta }^{\mathrm{\prime }}$ with $\beta <{\beta }^{\mathrm{\prime }}<\min \{1,k-m-\frac{n}{p}\}$. By [F3] the sequence is bounded in $C^{m,{\beta }^{\mathrm{\prime }}}(\bar{\mathrm{\Omega }})$, so each of its finitely many derivative families of orders at most $m$ is uniformly bounded and equicontinuous on the compact set $\bar{\mathrm{\Omega }}$. By [F6], applying Arzel`a--Ascoli successively to these derivative families gives a common subsequence on which every $D^{\alpha }{u}_{j_k}$ converges uniformly to a continuous function $v_{\alpha }$ on $\bar{\mathrm{\Omega }}$. On each ball compactly contained in $\mathrm{\Omega }$, [F7] applied to line segments shows that $v_{\alpha +e_i}$ is the classical $i$-th derivative of $v_{\alpha }$ whenever $\mathrm{\mid }\alpha \mathrm{\mid }<m$; hence $v_0$ is a representative in $C^m(\mathrm{\Omega })$ whose derivatives through order $m$ extend continuously to $\bar{\mathrm{\Omega }}$. For $\beta =0$ the uniform convergence is the desired $C^{m,0}$ convergence. For $\beta >0$, [F8] applied to each difference $D^{\alpha }(u_{j_k}-u_{j_{\mathrm{\ell }}})$, using uniform convergence and the uniform $C^{0,{\beta }^{\mathrm{\prime }}}$ bounds, gives convergence in $C^{m,\beta }(\bar{\mathrm{\Omega }})$ to $v_0$: each uniform limit $v_{\alpha }$ retains the bounded ${\beta }^{\mathrm{\prime }}$ seminorm by passage to the limit in the pointwise difference quotients, so [F8] applies directly to $D^{\alpha }{u}_{j_k}-v_{\alpha }$. Thus the asserted compact embedding holds, and the Axiom of Choice supplies the subsequence and the choice interfaces of [F2] and [F6].