Local $L^p$ compactness of $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}$-bounded sequences

If $\mathrm{\Omega }=\mathrm{\varnothing }$, the assertions are immediate. Otherwise enumerate the countable cover in [F1] as $(B_m{)}_{m\ge 1}$. The local boundedness hypothesis makes $(u_j)$ bounded in $W^{1,p}(B_1)$, so [F2] gives a subsequence converging in $L^p(B_1)$. Recursively, after obtaining a subsequence converging on $B_1,\dots ,B_m$, apply [F2] to that subsequence on $B_{m+1}$ and retain a further subsequence converging there. Countable and Dependent Choice select these nested subsequences. The diagonal sequence $(v_j)$, taking the $j$-th term of the $j$-th subsequence, is eventually a subsequence of each stage; hence it converges in $L^p(B_m)$ for every $m$.

Let $u_m$ be the $L^p(B_m)$ limit of $(v_j)$. On every overlap $B_m\cap B_{\mathrm{\ell }}$ the limits $u_m$ and $u_{\mathrm{\ell }}$ agree almost everywhere, by uniqueness of limits of the same sequence in $L^p(B_m\cap B_{\mathrm{\ell }})$. Since the cover is countable, these compatible classes patch to a class $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p(\mathrm{\Omega })$. If ${\mathrm{\Omega }}^{\mathrm{\prime }}⋐\mathrm{\Omega }$, compactness and [F5] give a finite subcover ${\mathrm{\Omega }}^{\mathrm{\prime }}\subseteq {\bigcup }_{m\in J}{B}_m$; therefore $\mathrm{\parallel }{v}_j-u{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}^{\mathrm{\prime }})}^p\le {\sum }_{m\in J}\mathrm{\parallel }{v}_j-u{\mathrm{\parallel }}_{L^p(B_m)}^p⟶0.$ Thus $v_j\to u$ in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p(\mathrm{\Omega })$.

Suppose $1<p<\mathrm{\infty }$ and fix $m$. The sequence $(v_j)$ is bounded in $W^{1,p}(B_m)$. By [F3], after finitely many further subsequence extractions, its function and each of its $n$ weak derivatives converge weakly in $L^p(B_m)$, say $v_{j_k}\rightharpoonup w$ and $D_i{v}_{j_k}\rightharpoonup w_i$. Step 2.1 gives strong convergence to $u_m$ in $L^p(B_m)$, so the weak limit is $w=u_m$. Passing to the limit in the weak-derivative identities against each $\phi \in C_c^{\mathrm{\infty }}(B_m)$ and using [F4] gives $u_m\in W^{1,p}(B_m)$ with $D_i{u}_m=w_i$. This argument may use a further subsequence depending on $m$: it identifies the already fixed strong limit $u_m$, so it identifies the derivatives of the already fixed limit on every ball without changing the diagonal sequence of step 1.1. On overlaps these derivative classes agree by the weak test identity. For any ${\mathrm{\Omega }}^{\mathrm{\prime }}⋐\mathrm{\Omega }$, choose a finite ball subcover of $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$ and an ambient smooth partition equal to one near that compact set (Finite ambient partitions near compact sets). Testing after multiplication by the partition pieces proves the weak derivative identity on ${\mathrm{\Omega }}^{\mathrm{\prime }}$; the partition-gradient terms sum to zero, and the finitely many local $L^p$ bounds give global $L^p({\mathrm{\Omega }}^{\mathrm{\prime }})$ bounds. Thus $u\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega })$. For $p=1$ this weak-compactness step is unavailable, and no membership of the limit in $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}$ is asserted. The assumed Axiom of Choice supplies the countable selections in step 1.1 and the Countable and Dependent Choice interfaces of [F2] and [F3].