Local weak scalar potentials and continuation on a simply connected open set

Let $B=B(a,r)⋐G$ and choose $r^{\mathrm{\prime }}>r$ with $B(a,r^{\mathrm{\prime }})⋐G$. Zero-extend $v{\mathrm{\mid }}_{B(a,r^{\mathrm{\prime }})}$ as an $L^2$ function and convolve at radii ${\epsilon }_j\downarrow 0$ smaller than $r^{\mathrm{\prime }}-r$. The resulting $v_j$ are smooth and curl free on a neighborhood of $B$, and $v_j\to v$ in $L^2(B)$ by [F2]. Put $g_j(x)={\int }_0^1{v}_j(a+t(x-a))\cdot (x-a)dt$. The smooth curl-free identity ${\mathrm{\partial }}_i{v}_{j,k}={\mathrm{\partial }}_k{v}_{j,i}$ gives ${\mathrm{\partial }}_i{g}_j={\int }_0^1\frac{d}{dt}(t{v}_{j,i}(a+t(x-a)))dt=v_{j,i}(x)$. Subtract $(g_j{)}_B$. For any $j,k$, [F1] bounds the $H^1$ norm of this normalized difference by $C_B\mathrm{\parallel }{v}_j-v_k{\mathrm{\parallel }}_{L^2(B)}$. Completeness gives a limit $g_B\in H^1(B)$ with mean zero and gradient $v$. No passage through smooth de Rham cohomology is made.

Cover $G$ by its countable rational balls with compact closure in $G$, and choose the primitives in step 1.1. Every nonempty intersection of two such balls is convex and connected. By [F3], $g_B-g_C=c_{BC}$ almost everywhere on $B\cap C$, with a unique real constant; $c_{BC}=-c_{CB}$, and $c_{BC}+c_{CD}=c_{BD}$ whenever the triple intersection is nonempty. Consider a germ of a primitive to mean a local $H^1$ primitive on a neighborhood of a point, identifying two when they agree almost everywhere on a smaller neighborhood. Fix one germ at a basepoint. Along a path, choose a finite subdivision whose successive subpaths lie in balls, and transport the current primitive by adding the unique constant needed to agree on an overlap at each subdivision point. This defines the terminal germ. Inserting extra subdivision points does not change it, and changing the covering balls gives the same result: refine around every subdivision point so that all balls compared contain that point, use their nonempty open intersections, and use the displayed cocycle identity. Thus transport is well defined along a path and reverses under reversal of that path.

Transport is invariant under endpoint-fixed path homotopy. For a continuous homotopy $H:[0,1{]}^2\to G$, compactness gives a finite subcover of its image by the ball cover and a Lebesgue number for the pullback cover of the square. A sufficiently fine rectangular grid then maps every closed cell into one ball. The transport around the boundary of that cell is the identity: compute it with that one ball's primitive; subdivision and ball independence from step 2.1 permit this computation. Compose these identities over all cells. Every interior edge is traversed once in each direction, so its transports cancel. The two vertical boundary paths are constant when endpoints are fixed; hence the transports along the two horizontal paths agree. If $G$ has the stated loop property, any two paths from the basepoint to a given point have equal terminal germs by applying this argument to their concatenation with one reversed. A connected open subset of Euclidean space is path connected: the set reachable by polygonal paths is both open and relatively closed, since a ball about any point joins that point to all nearby points. Thus there is a terminal germ at every point.

On a small ball around any point, path independence identifies these terminal germs with one fixed local primitive plus a constant. The resulting local functions therefore agree almost everywhere on overlaps. They define an $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$ class on $G$: enumerate the countable ball cover and use the first applicable representative on each measurable difference of balls; the overlap exceptional sets have countable union of measure zero. Any compact subset has a finite ball subcover, so the class is $H^1$ locally, and its weak derivative is $v$ there and hence globally on compactly supported tests. This is $f$. Any two choices differ by a locally $H^1$ zero-gradient function, so [F3] makes the difference constant. AC is used only for the cited analytic interfaces and the countable local representative/primitive choices.