Sobolev functions paste across an overlap

By [F11], AC gives Countable Choice, so the uniqueness interface [F4] applies. For $\mathrm{\mid }\alpha \mathrm{\mid }\le k$ the restricted classes $D^{\alpha }{u}_U{\mathrm{\mid }}_{U\cap V}$ and $D^{\alpha }{u}_V{\mathrm{\mid }}_{U\cap V}$ are both weak $\alpha$-derivatives of the common class $u_U=u_V$ on $U\cap V$: this is the restriction clause of [F4] applied in the open set $U\cap V$. Hence $D^{\alpha }{u}_U=D^{\alpha }{u}_V\text{almost everywhere on }U\cap V\mathrm{.}$ Define $g_{\alpha }$ on $\mathrm{\Omega }$ by $g_{\alpha }=D^{\alpha }{u}_U$ on $U$ and $g_{\alpha }=D^{\alpha }{u}_V$ on $V\setminus U$; the two defining pieces agree almost everywhere on their overlap, so $g_{\alpha }$ is a well-defined member of $L^p(\mathrm{\Omega };\mathbb{K})$ by [F5], [F6] and [F9]: it is measurable, its finite-$p$ integral is at most the sum of the two local $p$-th-power integrals, and its essential bound at $p=\mathrm{\infty }$ is at most the maximum of the two local bounds. Write $g_0=:u$; restricting the defining identity shows that $u$ agrees with $u_U$ almost everywhere on $U$ and with $u_V$ almost everywhere on $V$.

Fix a test $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and choose the finite grouped functions ${\eta }_{U,\phi },{\eta }_{V,\phi }$ of [F1]. Their supports lie in $U,V$, respectively, and $({\eta }_{U,\phi }+{\eta }_{V,\phi })\phi =\phi$ on $\mathrm{\Omega }$. For each $\mathrm{\mid }\alpha \mathrm{\mid }\le k$, the products ${\eta }_{U,\phi }\phi$ and ${\eta }_{V,\phi }\phi$ are tests supported in $U$ and $V$ by [F1] and [F10], so the weak identities of [F3] for $u_U$ on $U$ and $u_V$ on $V$ give ${\int }_U{u}_U{D}^{\alpha }({\eta }_{U,\phi }\phi )=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_U(D^{\alpha }{u}_U){\eta }_{U,\phi }\phi ,{\int }_V{u}_V{D}^{\alpha }({\eta }_{V,\phi }\phi )=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_V(D^{\alpha }{u}_V){\eta }_{V,\phi }\phi \mathrm{.}$ Since their sum times $\phi$ equals $\phi$, the derivatives satisfy $D^{\alpha }\phi =D^{\alpha }(({\eta }_{U,\phi }+{\eta }_{V,\phi })\phi )$ on $\mathrm{\Omega }$. Inserting the definitions of $u$ and $g_{\alpha }$ from step 1.1 and using [F6] and [F7] to replace the integrals over $U$ and $V$ by integrals over $\mathrm{\Omega }$, then adding the identities gives ${\int }_{\mathrm{\Omega }}u{D}^{\alpha }\phi dx=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{\mathrm{\Omega }}{g}_{\alpha }\phi dx\mathrm{.}$

The test $\phi$ in step 2.1 was arbitrary, so by [F3] each $g_{\alpha }$ is a weak $\alpha$-derivative of $u$ on $\mathrm{\Omega }$; since $g_{\alpha }\in L^p(\mathrm{\Omega })$ and $\mathrm{\mid }\alpha \mathrm{\mid }\le k$ was arbitrary, [F2] gives $u\in W^{k,p}(\mathrm{\Omega };\mathbb{K})$ with $D^{\alpha }u=g_{\alpha }$, restricting to $D^{\alpha }{u}_U$ on $U$ and $D^{\alpha }{u}_V$ on $V$ almost everywhere. If $\tilde{u}$ is another class with the same two restrictions, then $\tilde{u}=u$ almost everywhere on $U$ and on $V$, hence on $\mathrm{\Omega }=U\cup V$; so the pasted class is unique.

For the norm bound, fix $\mathrm{\mid }\alpha \mathrm{\mid }\le k$. For finite $p$, [F6] applied to $\mathrm{\mid }{g}_{\alpha }{\mathrm{\mid }}^p$ and the pointwise inequality ${\mathbf{1}}_{U\cup V}\le {\mathbf{1}}_U+{\mathbf{1}}_V$ give ${\int }_{\mathrm{\Omega }}\mathrm{\mid }{g}_{\alpha }{\mathrm{\mid }}^p\le {\int }_U\mathrm{\mid }{D}^{\alpha }{u}_U{\mathrm{\mid }}^p+{\int }_V\mathrm{\mid }{D}^{\alpha }{u}_V{\mathrm{\mid }}^p,$ and summing over the finitely many $\mathrm{\mid }\alpha \mathrm{\mid }\le k$ yields $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}^p\le \mathrm{\parallel }{u}_U{\mathrm{\parallel }}_{W^{k,p}(U)}^p+\mathrm{\parallel }{u}_V{\mathrm{\parallel }}_{W^{k,p}(V)}^p$ by [F2]. For $p=\mathrm{\infty }$, [F9] gives $\mathrm{\parallel }{g}_{\alpha }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le \max (\mathrm{\parallel }{D}^{\alpha }{u}_U{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(U)},\mathrm{\parallel }{D}^{\alpha }{u}_V{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(V)})$, and taking the maximum over the finitely many $\mathrm{\mid }\alpha \mathrm{\mid }\le k$ gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,\mathrm{\infty }}(\mathrm{\Omega })}\le \mathrm{\parallel }{u}_U{\mathrm{\parallel }}_{W^{k,\mathrm{\infty }}(U)}+\mathrm{\parallel }{u}_V{\mathrm{\parallel }}_{W^{k,\mathrm{\infty }}(V)}$ by [F2].

The interfaces [F8] and [F6] also show that every integrand above is integrable: the test derivatives are bounded with compact support, so they lie in every $L^{p^{\mathrm{\prime }}}$ on the relevant pieces, and the weak derivative classes lie in $L^p$. The bound is special to the two-set cover: the inequality used ${\mathbf{1}}_{U\cup V}\le {\mathbf{1}}_U+{\mathbf{1}}_V$, whose analogue for an infinite cover would require summability. If $\mathrm{\Omega }=\mathrm{\varnothing }$, then $U=V=\mathrm{\varnothing }$ and all classes are the zero class, so the identities are trivial. The Axiom of Choice is used only through [F11] to obtain Countable Choice for the uniqueness interface [F4] and the Sobolev and integral interfaces [F2], [F7], [F8]; the locally finite partition in [F1] is choice-free. $\mathrm{\square }$