Discrete de Rham complexes and edge-element foundations

Put $M=\bar{\mathrm{\Omega }}$. It is compact by boundedness and closedness. Near each boundary point a Lipschitz graph chart $x_3=\varphi (x^{\mathrm{\prime }})$ straightens $M$ by the explicit homeomorphism $(x^{\mathrm{\prime }},x_3)\mapsto (x^{\mathrm{\prime }},x_3-\varphi (x^{\mathrm{\prime }}))$ to a closed halfspace neighborhood. Interior points use Euclidean balls. These charts show that $M$ is a Hausdorff second-countable three-manifold with topological boundary exactly $\mathrm{\partial }\mathrm{\Omega }$ and interior exactly $\mathrm{\Omega }$. Its orientation is the standard Euclidean orientation on its interior; this is precisely the orientation convention for manifolds with boundary in [F2]. By [F1], $\mathrm{\Omega }\hookrightarrow M$ is a homotopy equivalence. Contractibility of $\mathrm{\Omega }$, [F3], and factorization through a point give $H^2(M;\mathbb{R})=0$: the point has one singular simplex in each degree, and its real cochain differentials are successively zero and identity, so its positive cohomology is zero. Also $M$ is path connected: its interior is path connected by contractibility, and a half-chart path joins any boundary point to its interior. The augmentation identifies $H_0(M;\mathbb{R})$ with $\mathbb{R}$, because any difference of points is the boundary of a path and every zero-chain of coefficient sum zero is a finite linear combination of such differences. The boundary is nonempty: a ray from any interior point eventually leaves the bounded set and its first exit point is a boundary point.

By [F2], $H_1(M,\mathrm{\partial }M;\mathbb{R})=0$. The exact sequence in [F3] then makes $H_0(\mathrm{\partial }M;\mathbb{R})\to H_0(M;\mathbb{R})$ injective. The boundary is a two-manifold by the graph charts, hence locally path connected, so its connected components are path components. If there were two, choose one point in each. Their difference is nonzero in $H_0(\mathrm{\partial }M;\mathbb{R})$: the sum of coefficients on the first path component defines a linear functional on zero-chains which kills every one-simplex boundary, yet sends this difference to one. Its image in $H_0(M;\mathbb{R})$ is zero since the two points are joined by a path in $M$, contradicting injectivity. Therefore the nonempty boundary is connected. No finite-element incidence matrix is used to infer this continuum topological claim.

It suffices first to treat the special graph $G=\{(x^{\mathrm{\prime }},x_3):x_3>\varphi (x^{\mathrm{\prime }})\}$, with $\varphi$ globally $L$-Lipschitz. A local graph extends to a global graph without increasing $L$ by $\varphi (x^{\mathrm{\prime }})={\inf }_{y^{\mathrm{\prime }}}(\varphi (y^{\mathrm{\prime }})+L\mathrm{\mid }{x}^{\mathrm{\prime }}-y^{\mathrm{\prime }}\mathrm{\mid })$, taking $y^{\mathrm{\prime }}$ over the original graph base; for $L=0$ use the same constant. Localized input will have support strictly inside the original chart. The construction for $G$ is truncated at side length one, which fixes an inhomogeneous length scale. Rotations, translations, and a fixed dilation transfer it to any chart scale.

Decompose $G^c\setminus \mathrm{\partial }G$ into maximal dyadic cubes $Q$ for which $20\sqrt{3}{\mathrm{\ell }}_Q\le \mathrm{dist}(Q,\mathrm{\partial }G)$. Cubes of sufficiently small size containing any given exterior point satisfy this condition, while large ancestors fail, so maximal cubes exist and cover the exterior, with disjoint interiors. Failure for the parent and the triangle inequality give $20\sqrt{3}{\mathrm{\ell }}_Q\le \mathrm{dist}(Q,\mathrm{\partial }G)\le C{\mathrm{\ell }}_Q$. For two cubes whose $9\mathrm{/}8$ enlargements meet, the distance function is 1-Lipschitz; these two inequalities imply $C_0^{-1}{\mathrm{\ell }}_Q\le {\mathrm{\ell }}_R\le C_0{\mathrm{\ell }}_Q$, with a dimension-only $C_0$, and boundedly many such neighbors by volume packing. The enlarged cubes have bounded overlap for the same reason. Pick a fixed nonnegative smooth bump supported in the $9\mathrm{/}8$ enlargement of the unit cube and positive on the closed unit cube; rescale it to ${\eta }_Q$ and put ${\theta }_Q={\eta }_Q\mathrm{/}({\sum }_R{\eta }_R)$. The denominator is bounded below, only boundedly many comparable scales meet, and differentiation of this quotient gives $\mathrm{\mid }{D}^j{\theta }_Q\mathrm{\mid }\le C_j{\mathrm{\ell }}_Q^{-j}$ for $j=0,1,2$. Thus $({\theta }_Q)$ is a locally finite smooth partition on the exterior.

Restrictions of smooth functions defined in a neighborhood of $\bar{G}$ are dense in $H^m(G)$ for $m=1,2$. First cut off at large radius, with convergence in $H^m$ by the product rule and integrable tails. For the resulting compactly supported $f$, translate inward: $f_t(x)=f(x+t{e}_3)$ is defined on the neighborhood $G-t{e}_3$ of $\bar{G}$. Each derivative on $G$ is the restriction of the corresponding derivative of $f$, extended by zero merely as an $L^2$ function and translated; [F2] therefore gives convergence to $D^{\alpha }f$ in $L^2(G)$ as $t\downarrow 0$. Convolve $f_t$ with a mollifier of radius $\epsilon <t\mathrm{/}(2(1+L))$. Every ball used at a point of $\bar{G}$ stays inside $G-t{e}_3$, so differentiation under the convolution is legitimate there. [F2] gives convergence of all these derivatives, by choosing $\epsilon$ sufficiently small after $t$. Fixed cutoffs retain compact support. This proves the asserted density without assuming an extension theorem.

Write ${\delta }_Q=\varphi (a_Q^{\mathrm{\prime }})-a_{Q,3}$. Distance to a Lipschitz graph and vertical distance satisfy $\mathrm{dist}(a_Q,\mathrm{\partial }G)\le {\delta }_Q\le \sqrt{1+L^2}\mathrm{dist}(a_Q,\mathrm{\partial }G)$: the second inequality follows by comparing to any graph point and applying Cauchy–Schwarz to $L\mathrm{\mid }{a}_Q^{\mathrm{\prime }}-b^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{\mid }{a}_{Q,3}-\varphi (b^{\mathrm{\prime }})\mathrm{\mid }$. Hence $c{\mathrm{\ell }}_Q\le {\delta }_Q\le C_L{\mathrm{\ell }}_Q$. Fix $A>2C_L+100(1+L)$ and let $Q^{\ast }$ be the cube of side ${\mathrm{\ell }}_Q$ centered at $b_Q=a_Q+A{\mathrm{\ell }}_Q{e}_3$. It lies in $G$, its distance to the graph is between $c_L{\mathrm{\ell }}_Q$ and $C_L^{\mathrm{\prime }}{\mathrm{\ell }}_Q$, and $\mathrm{dist}(Q,Q^{\ast })\le A{\mathrm{\ell }}_Q$. The cubes $Q^{\ast }$ have bounded overlap: at any point in one of them their scale is comparable to its distance from the graph, and their original disjoint cubes lie in a ball of radius $C_L{\mathrm{\ell }}_Q$.

For neighboring exterior cubes $Q,R$, their interior cubes $Q^{\ast },R^{\ast }$ can be joined by a chain of a bounded number (depending only on $L$) of interior cubes of sides comparable to ${\mathrm{\ell }}_Q$, with consecutive overlaps containing a cube of side $c_L{\mathrm{\ell }}_Q$. Here is an explicit construction. Join $b_Q$ vertically upward to height $H=\max (b_{Q,3},b_{R,3})+L\mathrm{\mid }{b}_Q^{\mathrm{\prime }}-b_R^{\mathrm{\prime }}\mathrm{\mid }+10\sqrt{3}\max ({\mathrm{\ell }}_Q,{\mathrm{\ell }}_R)$, then horizontally to the vertical line through $b_R$, then vertically down to $b_R$. Each segment stays above the graph by at least $c_L{\mathrm{\ell }}_Q$, has length at most $C_L{\mathrm{\ell }}_Q$, and stays at distance at most $C_L{\mathrm{\ell }}_Q$ from the graph. Subdivide these segments into steps of length $b{\mathrm{\ell }}_Q\mathrm{/}4$, and use centered cubes of side $b{\mathrm{\ell }}_Q$, with $b>0$ small enough to lie in $G$; append $Q^{\ast },R^{\ast }$ as endpoints. Call the union $B_{QR}$. These bridges have bounded overlap when all neighbor pairs are considered: every point in a bridge has graph distance comparable to its scale, and all corresponding exterior cubes are disjoint cubes of comparable size in a fixed multiple of a ball of that scale. There are boundedly many of them by volume packing.

For ${\mathrm{\ell }}_Q\le 1$ put $P_{Q}f(x)=f_{Q^{\ast }}+(\mathrm{\nabla }f{)}_{Q^{\ast }}\cdot (x-b_Q)$, an affine polynomial determined linearly by the $H^1$ class. Put $P_Q=0$ when ${\mathrm{\ell }}_Q>1$. For an interior cube $S$ of side $s$ and center $c_S$, write $P_{S}f=f_S+(\mathrm{\nabla }f{)}_S\cdot (x-c_S)$. Applying [F1] first to $\mathrm{\nabla }f$ and then to $f-P_{S}f$, whose mean is zero, gives $\mathrm{\parallel }f-P_{S}f{\mathrm{\parallel }}_{L^2(S)}\le C{s}^2\mathrm{\parallel }{D}^{2}f{\mathrm{\parallel }}_{L^2(S)}$ and $\mathrm{\parallel }\mathrm{\nabla }(f-P_{S}f){\mathrm{\parallel }}_{L^2(S)}\le Cs\mathrm{\parallel }{D}^{2}f{\mathrm{\parallel }}_{L^2(S)}$ for $H^2$ data. For merely $H^1$ data, $\mathrm{\parallel }f-P_{S}f{\mathrm{\parallel }}_{L^2(S)}\le Cs\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_{L^2(S)}$. These estimates also hold on the finite cube chains in step 4.1, with constants depending on their uniform chain geometry. To see the latter assertion without assuming a chain Poincare theorem, compare the polynomials on each consecutive overlap. For consecutive chain cubes $S,T$, choose a cube $O\subset S\cap T$ of side $\rho \asymp s$. On $O$ the triangle inequality gives $\mathrm{\parallel }{P}_{S}f-P_{T}f{\mathrm{\parallel }}_{L^2(O)}\le \mathrm{\parallel }f-P_{S}f{\mathrm{\parallel }}_{L^2(S)}+\mathrm{\parallel }f-P_{T}f{\mathrm{\parallel }}_{L^2(T)}$, so the already proved residual bounds apply on the actual cubes $S,T$, without assuming their union lies in another interior cube. If an affine polynomial is $P(x)=a+b\cdot (x-c_O)$ on $O$, direct integration gives $\mathrm{\parallel }P{\mathrm{\parallel }}_{L^2(O)}^2={\rho }^3\mathrm{\mid }a{\mathrm{\mid }}^2+{\rho }^5\mathrm{\mid }b{\mathrm{\mid }}^2\mathrm{/}12$. Since the exterior target cube $Q$ has side comparable to $s$ and lies within distance $C_{L}s$ of each chain cube, this formula gives $\mathrm{\parallel }{D}^{j}P{\mathrm{\parallel }}_{L^2(Q)}\le C_L{s}^{-j}\mathrm{\parallel }P{\mathrm{\parallel }}_{L^2(O)}$ for $j=0,1$, while $D^{2}P=0$. Apply it to $P_{S}f-P_{T}f$ and telescope along the chain. There are boundedly many links, so summing those finitely many bounds proves, for $Q,R$ neighbors with both scales at most one, $\mathrm{\parallel }{D}^j(P_{Q}f-P_{R}f){\mathrm{\parallel }}_{L^2(Q)}\le C{\mathrm{\ell }}_Q^{m-j}\mathrm{\parallel }{D}^{m}f{\mathrm{\parallel }}_{L^2(B_{QR})},0\le j\le m,m=1,2.$ Here $D^2$ of an affine polynomial is zero; $B_{QR}$ may be enlarged by the finitely many overlapping chain cubes used in the comparison, retaining step 4.1's overlap property.

Define $Ef=f$ on $G$ and $Ef={\sum }_Q{\theta }_Q{P}_{Q}f$ on its exterior. For an exterior cube $Q$, subtract $P_Q$ inside the sum and differentiate. For second derivatives, all terms containing $D^2{P}_Q$ vanish, and derivatives of the partition of total order one or two multiply derivatives of $P_R-P_Q$. Step 5.1 bounds their $L^2(Q)$ norm by $C{\sum }_{R\sim Q}\mathrm{\parallel }{D}^{2}f{\mathrm{\parallel }}_{L^2(B_{QR})}$. For first derivatives, the remaining term $\mathrm{\nabla }{P}_Q$ is bounded by the average of $\mathrm{\nabla }f$ on $Q^{\ast }$, and the residual terms are bounded by the $m=1$ version of step 5.1. For the zeroth derivative, $\mathrm{\parallel }{P}_Q{\mathrm{\parallel }}_{L^2(Q)}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(Q^{\ast })}+{\mathrm{\ell }}_Q\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_{L^2(Q^{\ast })})$. Cubes with ${\mathrm{\ell }}_Q>1$ have zero polynomial; if a neighbor is nonzero, all scales involved lie between $C_0^{-1}$ and $C_0$, so one bounds its polynomial and both its derivatives directly by $C\mathrm{\parallel }f{\mathrm{\parallel }}_{H^m(R^{\ast })}$ instead of a difference estimate. Summing squares, using bounded neighbor counts and the bounded overlap of interior cubes and bridges, proves $\mathrm{\parallel }Ef{\mathrm{\parallel }}_{H^m(G)}+\mathrm{\parallel }Ef{\mathrm{\parallel }}_{H^m({\mathbb{R}}^3\setminus \bar{G})}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{H^m(G)},m=1,2,$ where the norms on the right are taken only when defined. This estimates derivatives separately on the two open sides; the next steps verify that there is no boundary distribution.

For such smooth $f$, use a fixed ambient smooth continuation near the graph to compare with $Ef$. At an exterior point $x$ at distance $d$ from the graph, every active cube has ${\mathrm{\ell }}_Q\asymp d$ and $Q^{\ast }$ is within distance $C_{L}d$ of $x$. Taylor's formula and the affine reproduction of $P_Q$ therefore give $\mathrm{\mid }Ef(x)-f(x)\mathrm{\mid }\le C_f{d}^2$ and $\mathrm{\mid }\mathrm{\nabla }(Ef-f)(x)\mathrm{\mid }\le C_{f}d$ on compact neighborhoods of the graph. To verify the weak derivatives across the graph, define on the exterior $\rho (x)={\sum }_Q{\theta }_Q(x){\mathrm{\ell }}_Q$. Local finiteness makes $\rho$ smooth there. At each exterior point the active cube sizes are comparable to $d$, so $cd\le \rho \le Cd$. Since ${\sum }_Q{\theta }_Q=1$, differentiating after subtracting one active scale gives $\mathrm{\mid }\mathrm{\nabla }\rho \mathrm{\mid }\le C$ and $\mathrm{\mid }{D}^2\rho \mathrm{\mid }\le C\mathrm{/}d$, using $\mathrm{\mid }{D}^j{\theta }_Q\mathrm{\mid }\le C_j{\mathrm{\ell }}_Q^{-j}$ and the bounded number of active cubes. Choose a smooth scalar cutoff $\zeta$ that is zero on $(-\mathrm{\infty },1]$ and one on $[2,\mathrm{\infty })$, and put ${\chi }_t(x)=\zeta (\rho (x)\mathrm{/}t)$ outside, extended by zero across $\mathrm{\partial }G$. This is smooth across the boundary because it vanishes in a full exterior collar. On its transition strip, $d\asymp t$ and the chain rule gives $\mathrm{\mid }D{\chi }_t\mathrm{\mid }\le C\mathrm{/}t$ and $\mathrm{\mid }{D}^2{\chi }_t\mathrm{\mid }\le C\mathrm{/}{t}^2$. In a graph chart, the vertical gap $\delta =\mathrm{\mid }{x}_3-\varphi (x^{\mathrm{\prime }})\mathrm{\mid }$ satisfies $\delta \le \sqrt{1+L^2}d$: compare $x$ to any graph point and use the Lipschitz bound on $\varphi$, then take the infimum. Thus on a compact chart the strip $d<Ct$ has vertical sections of length at most $C^{\mathrm{\prime }}t$, so its volume is $O(t)$ by [F5]. The product-rule errors from $h=Ef-f$ on that strip satisfy $\mathrm{\mid }h\mathrm{\mid }=O(t^2)$ and $\mathrm{\mid }Dh\mathrm{\mid }=O(t)$, so $hD{\chi }_t$, $DhD{\chi }_t$, and $h{D}^2{\chi }_t$ tend to zero in $L^2$; the omitted piecewise second derivatives tend to zero in $L^2$ by absolute continuity of their squared integrals. Thus the cutoffs converge in $H^2$ locally and prove that the zero-inside function has precisely the piecewise weak derivatives, with no boundary distribution. The same argument with only first derivatives proves the $H^1$ assertion. Consequently step 6.1 is a genuine whole-space Sobolev bound for the smooth dense class.

For arbitrary $H^m$ data take the smooth approximating sequence in step 2.2. The bound in step 6.1 and step 7.1 makes its extensions Cauchy in the complete whole-space $H^m$ norm. Their limit equals the original input on $G$. On each exterior cube its formula equals the displayed averaged-polynomial formula, since the finitely many local averages and derivatives entering that formula converge. Thus the $H^1$ and $H^2$ extensions are the same operator on their common domain, rather than two incompatible choices. This completes the special graph case.

For bounded $D$, cover its compact boundary by finitely many graph charts and choose smooth cutoffs supported strictly inside them, together with an interior cutoff, summing to one near $\bar{D}$. Each localized function is extended by zero within its special graph outside its original chart; since its cutoff vanishes near the chart sides, this is an $H^m$ function on the graph with the expected product derivatives. Apply the graph operator, then multiply the result by a second fixed cutoff supported in the chart and equal to one on the first cutoff's support. On $D$ the result is the localized original function, including the part where that function is zero. The interior term is extended by zero, with no interface near its support. Summing these finitely many extensions gives $E$, preserves the restriction identity, and gives both asserted boundedness estimates by the weak product rule. A final cutoff equal to one near $\bar{D}$ confines support. Finite graph covers work equally for disconnected $D$; no connectivity or topology is used. All construction coefficients are fixed independently of $f$. This proves the statement.

Fix the shape-regularity bound $\sigma$. For a cell $K$, let $F_K(\widehat{x})=b_K+A_K\widehat{x}$ be its orientation-preserving map from the reference tetrahedron. Its three columns are edge vectors, so $\mathrm{\parallel }{A}_K\mathrm{\parallel }\le \sqrt{3}{h}_K$. Since $K$ contains an inscribed ball of radius ${\rho }_K\ge h_K\mathrm{/}\sigma$ and $\mathrm{\mid }K\mathrm{\mid }=\mathrm{\mid }\det A_K\mathrm{\mid }\mathrm{/}6$, we have $\mathrm{\mid }\det A_K\mathrm{\mid }\ge 8\pi (h_K\mathrm{/}\sigma {)}^3$. If $s_1\ge s_2\ge s_3$ are the singular values of $A_K$, then $s_1,s_2\le \sqrt{3}{h}_K$ and $s_1{s}_2{s}_3=\mathrm{\mid }\det A_K\mathrm{\mid }$, hence $s_3\ge c_{\sigma }{h}_K$. Thus every singular value is comparable to $h_K$. Every physical edge is the image under $A_K$ of one of the six fixed reference edge vectors, whose lengths are either $1$ or $\sqrt{2}$; consequently all edge lengths lie between $c_{\sigma }{h}_K$ and $C{h}_K$. At a vertex of $K$, its three emanating edges form a matrix $E=A_K{R}_i$, where $R_i$ is one of four fixed invertible reference matrices. The preceding singular-value bounds therefore give $s_{\min }(E)\ge c_{\sigma }{h}_K$. Divide each column by its edge length to obtain $B$ with unit columns. Since every edge length is at most $h_K$, $s_{\min }(B)\ge a>0$ uniformly; decrease $a$ to at most one. Also $\mathrm{\parallel }B\mathrm{\parallel }\le \sqrt{3}$. Put $u_0=(1\mathrm{/}6,1\mathrm{/}6,1\mathrm{/}6)$ and $p=B{u}_0$. Then $\mathrm{\mid }p\mathrm{\mid }\le 1\mathrm{/}2$. If $\mathrm{\mid }z-p\mathrm{\mid }<a\mathrm{/}12$, then $\mathrm{\mid }{B}^{-1}z-u_0\mathrm{\mid }<1\mathrm{/}12$, so all coordinates of $B^{-1}z$ are positive. Consequently the open ball $B(p,a\mathrm{/}12)$ is contained in the open vertex cone $\{Bu:u_j>0\}$ and in the unit ball, since $\mathrm{\mid }z\mathrm{\mid }<1\mathrm{/}2+a\mathrm{/}12<1$. The open cones of distinct cells at a fixed vertex are disjoint: a point in two such cones would give a sufficiently short ray from that vertex inside both cell interiors, contrary to the matching mesh. Thus their radius-$a\mathrm{/}12$ balls are disjoint and their centers are separated by at least $a\mathrm{/}6$ (otherwise their midpoint lies in both balls). All centers lie in $[-1,1{]}^3$. Fix an integer $M>12\sqrt{3}\mathrm{/}a$ and partition this cube by the coordinate grid into $M^3$ boxes of side $2\mathrm{/}M$, assigning points on division planes to one adjacent box by a fixed half-open convention and including the outer endpoints. Each box has diameter $2\sqrt{3}\mathrm{/}M<a\mathrm{/}6$, so contains at most one center. Hence at most $M^3$ cells meet any vertex. This is a finite grid and pigeonhole argument, with no surface-area formula, measure theorem or choice principle. Their link is connected through faces containing that vertex: in an interior star the link is a triangulated sphere, and in a boundary star it is a triangulated disk, since a Lipschitz graph has a ball or half-ball neighborhood. Across a shared face the two diameters are comparable because each face edge is comparable to each incident cell diameter. A face path in the link has bounded length, so every two cells meeting a vertex have comparable diameters, with a constant depending only on shape regularity. Assign $g_h(v)={\min }_{K\ni v}{h}_K$ at every vertex and interpolate affinely. Barycentric differentiation and $\mathrm{\mid }\mathrm{\nabla }{\lambda }_v\mathrm{\mid }\le C\mathrm{/}{h}_K$ give the asserted cellwise comparability and gradient bound.

In a boundary graph box choose its upward unit direction $e_j$. Shrink its support neighborhood so that every planar boundary piece meeting that neighborhood has outward normal $n$ satisfying $n\cdot e_j\le -{\alpha }_j<0$. This holds because for a graph $x_3=f(x^{\mathrm{\prime }})$, with the domain above it, every outward unit normal is $(\mathrm{\nabla }f,-1)\mathrm{/}(1+\mathrm{\mid }\mathrm{\nabla }f{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. Refine the finite boundary cover by finitely many concentric ball pairs whose larger closures lie in the assigned graph boxes and whose smaller balls still cover the compact boundary. Use [F1] to put a bump ${\beta }_j$ in each larger ball, equal to one on its smaller ball. On the open collar where $S={\sum }_j{\beta }_j>0$, put ${\chi }_j={\beta }_j\mathrm{/}S$ and $b_0={\sum }_j{\chi }_j{e}_j$. A compact-set cutoff $\eta$ supported in that collar and equal to one on a smaller collar gives the global smooth field $b=\eta b_0$, extended by zero away from the support of $\eta$. This is an explicitly normalized finite partition; no arbitrary-index partition theorem or choice axiom is imported. At each collar point every participating $e_j$ has the strict inward inequality for every nearby boundary face, so their convex combination does too, uniformly. After shrinking the collar, there is $r>0$ such that $b(x)+ry$ is strictly inward for all nearby faces and $-b(x)+ry$ strictly outward, for every $\mathrm{\mid }y\mathrm{\mid }\le 1$.

A bounded connected Lipschitz polyhedron is quasiconvex: close points in one boundary graph box can be joined by two vertical segments and a horizontal segment above the graph, of total length at most a fixed multiple of their distance; interior balls have the same property. A finite smaller cover gives this assertion for sufficiently close points. For remaining pairs, join the finitely many box centers through fixed interior polygonal paths and divide their bounded length by the fixed positive distance threshold. Componentwise the same argument applies to a disconnected domain, and distinct components have positive separation. Integrating the cellwise gradient along these paths gives the mesh-independent Euclidean Lipschitz bound for $g_h$. Extend it to a neighborhood by the explicit formula ${\tilde{g}}_h(x)={\inf }_{z\in \bar{\mathrm{\Omega }}}(g_h(z)+L\mathrm{\mid }x-z\mathrm{\mid })$, then truncate above by $\max g_h$; this is positive and $L$-Lipschitz and agrees on $\bar{\mathrm{\Omega }}$.

Here are the distance assertions, including concave corners. In any one graph box write $a=b(x)+ry$. Along the horizontal segment from $x^{\mathrm{\prime }}$ to $x^{\mathrm{\prime }}+\tau a^{\mathrm{\prime }}$, the graph is piecewise affine; on each piece the preceding normal inequality gives $a_3-\mathrm{\nabla }f\cdot a^{\mathrm{\prime }}\ge \beta >0$. Integration along that segment gives $f(x^{\mathrm{\prime }}+\tau a^{\mathrm{\prime }})\le f(x^{\mathrm{\prime }})+\tau a_3-\beta \tau$. Thus for $x\in \bar{\mathrm{\Omega }}$ the point $x+\tau a$ lies above the graph by at least $\beta \tau$, and its distance to the graph is at least $\beta \tau \mathrm{/}(1+L^2{)}^{1\mathrm{/}2}$. Replacing $b$ by $-b$ reverses the inequality and puts boundary points strictly below the graph. The finite cover supplies uniform constants and a uniform allowed $\tau$. Away from the collar, small displacements remain inside. Substitute $\tau =t{g}_h(x)$; the upper bound on $g_h$ is at most the domain diameter. Thus one fixed ${ϵ}_0$ works for every mesh.

Let ${\omega }_K$ be the union of cells sharing a vertex with $K$. The sum ${\psi }_K$ of the four global nodal hat functions for the vertices of $K$ is one on $K$, zero outside ${\omega }_K$, and has cellwise gradient at most $C\mathrm{/}{h}_K$: every cell where it is nonzero shares a vertex with $K$, so step 1.1 applies. The path argument of step 2.1 gives $\mathrm{Lip}({\psi }_K)\le C_{\mathrm{\Omega }}\mathrm{/}{h}_K$. Hence $\mathrm{dist}(K,\mathrm{\Omega }\setminus {\omega }_K)\ge c_{\mathrm{\Omega }}{h}_K$. The number of cells in each patch and the number of patches containing one cell are bounded by the vertex-incidence bound. This proves the claimed neighborhood and overlap properties for sufficiently small displacements, including on graded meshes.

The displacement field $g_h(\pm b+ry)$ has a uniformly bounded Lipschitz constant, since $g_h$, $b$ and their Lipschitz constants are uniformly bounded. Reduce ${ϵ}_0$ so this constant times ${ϵ}_0$ is below $1\mathrm{/}2$. Then $\frac{1}{2}\mathrm{\mid }x-z\mathrm{\mid }\le \mathrm{\mid }{F}_{t,y}^{\pm }(x)-F_{t,y}^{\pm }(z)\mathrm{\mid }\le \frac{3}{2}\mathrm{\mid }x-z\mathrm{\mid }$. The displacement is at most $Ct{h}_K$ on $K$. Step 3.1 now proves the patch assertion, and step 2.2 proves the interior and exterior boundary assertions.

Since $U$ is supported in $A$, Holder gives $\mathrm{\parallel }U{\mathrm{\parallel }}_1\le \mathrm{\mid }A{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }U{\mathrm{\parallel }}_2$, and therefore $\mathrm{\mid }\widehat{U}(\xi )\mathrm{\mid }\le \mathrm{\mid }A{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }U{\mathrm{\parallel }}_2$. For $\xi \mathrm{\ne }0$ define $\widehat{p}(\xi )=\frac{\xi \cdot \widehat{U}(\xi )}{2\pi i\mathrm{\mid }\xi {\mathrm{\mid }}^2},\widehat{w}(\xi )=\widehat{U}(\xi )-\frac{\xi (\xi \cdot \widehat{U}(\xi ))}{\mathrm{\mid }\xi {\mathrm{\mid }}^2}\mathrm{.}$ Set both values at zero arbitrarily, for instance zero. The multiplier defining $\widehat{w}$ is an orthogonal projection in ${\mathbb{C}}^3$, so $\mathrm{\parallel }\widehat{w}{\mathrm{\parallel }}_2\le \mathrm{\parallel }\widehat{U}{\mathrm{\parallel }}_2$, and $2\pi i\xi \widehat{p}$ is the complementary orthogonal projection of $\widehat{U}$. At $\mathrm{\mid }\xi \mathrm{\mid }<1$, $\mathrm{\mid }\widehat{p}{\mathrm{\mid }}^2\le (4{\pi }^2{)}^{-1}\mathrm{\mid }A\mathrm{\mid }\mathrm{\parallel }U{\mathrm{\parallel }}_2^2\mathrm{\mid }\xi {\mathrm{\mid }}^{-2}$, whose integral is finite in dimension three; at $\mathrm{\mid }\xi \mathrm{\mid }\ge 1$ it is bounded by $(4{\pi }^2{)}^{-1}\mathrm{\mid }\widehat{U}{\mathrm{\mid }}^2$. Thus $p\in H^1$ and $\mathrm{\parallel }p{\mathrm{\parallel }}_{H^1}\le C_A\mathrm{\parallel }U{\mathrm{\parallel }}_2$ by [F1]. This is the essential low-frequency check; an arbitrary nonsupported $L^2$ field would not give it.

The vector identity $\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{\mid }\widehat{w}{\mathrm{\mid }}^2=\mathrm{\mid }\xi \times \widehat{U}{\mathrm{\mid }}^2$ and the multiplier identity for curl give $\int 4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{\mid }\widehat{w}{\mathrm{\mid }}^2=\mathrm{\parallel }\mathrm{curl}U{\mathrm{\parallel }}_2^2$. Hence $w\in H^1$, with the asserted derivative norm. The displayed Fourier definitions give $U=w+\mathrm{\nabla }p$, $\xi \cdot \widehat{w}=0$, and $\xi \times \widehat{w}=\xi \times \widehat{U}$, so all asserted distributional identities follow by Fourier injectivity. Finally $\widehat{U}(-\xi )=\overline{\widehat{U}(\xi )}$ for real $U$. The two definitions have the corresponding conjugate symmetry (including the sign of $i$ in $\widehat{p}$); uniqueness of the inverse transform therefore gives real $p,w$. Every operation is linear, and [F1] supplies all norm claims.

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.