Compact-domain scalar Poisson solvability and fourth-order estimates

First establish pullback bounds independently of any equation or manifold Sobolev construction. Let $T$ be a smooth coordinate transition between interior patches or between the open interiors of matched boundary half-patches. Restrict to patches with compact closure inside the ambient chart neighbourhoods, allowing that closure to meet the physical boundary. Boundary-chart compatibility in [F1] gives local ambient smooth extensions of the transition and its inverse. After finite subdivision, their derivatives through any fixed integer order $m\ge 1$ are bounded and their Jacobians are bounded away from zero on these closures. For arbitrary $z\in H^m$ on the target patch, apply [F6] on compactly contained open interior subpatches: $D^{\nu }(z\circ T)$ is the sum of $(D^{\beta }z)\circ T$ times the prescribed bounded polynomials in derivatives of $T$, including order zero. These weak identities agree on overlapping interior subpatches. Exhaust the source open patch by such subpatches, choosing larger target interior subpatches containing their images. The norm estimates in [F6] have one common constant, since they depend only on $m$, dimension and the stated uniform chart and inverse bounds. Their right sides are bounded by the norm on the whole target patch. Taking the increasing integrals of the nonnegative squared derivative fields proves $\mathrm{\parallel }z\circ T{\mathrm{\parallel }}_{H^m}\le C\mathrm{\parallel }z{\mathrm{\parallel }}_{H^m}$ on the whole source half-patch, with the same weak identities. Every test function has compact support in that open half-patch, so these are its global weak derivatives. The inverse transition has the analogous bound. In particular $m=1$ applies to arbitrary functions and Cauchy sequences, without assuming that they are weak solutions; $m\le 4$ also supplies the higher-order pullbacks used below.

Let $d{\mu }_h$ be the measure of [F2], whose chart density is $\sqrt{\det h}dx$. Choose finitely many relatively compact coordinate patches and smooth cutoffs ${\chi }_i$ with ${\sum }_i{\chi }_i=1$. Each cutoff is supported away from artificial chart edges. In a boundary chart use a bounded smooth auxiliary domain in $\{x_3>0\}$ with a flat boundary portion covering that support; round its other boundary away from the support. Define $H^1(M)$ to consist of $L^2(d{\mu }_h)$ classes with local weak first derivatives and finite norm $\mathrm{\parallel }v{\mathrm{\parallel }}_{H_h^1}^2={\int }_M(v^2+\mathrm{\mid }dv{\mathrm{\mid }}_h^2)d{\mu }_h$. On the compact supports, smooth metric positivity and bounded chart Jacobians make this norm equivalent to the square root of the sum of the squared Euclidean $W^{1,2}$ norms of ${\chi }_{i}v$, extended by zero only past artificial edges. The product formula bounds these local norms by the intrinsic norm; conversely $v={\sum }_i{\chi }_{i}v$ and $dv={\sum }_{i}d({\chi }_{i}v)$ give the reverse bound. For arbitrary functions, coordinate changes obey the weak chain rule and bounded pullback estimates of step 1.1. Thus different such finite atlases give the same space and equivalent norms. All integrals are intrinsic density integrals, including on a nonorientable $M$.

If $v_j$ is Cauchy in this norm, each ${\chi }_i{v}_j$ converges in Euclidean $W^{1,2}$ by completeness. The local limits agree as pieces of one function: on overlaps the identities ${\chi }_l({\chi }_i{v}_j)={\chi }_i({\chi }_l{v}_j)$ pass to the $L^2$ limits using bounded coordinate pullbacks. Sum the finitely many pieces to obtain $v$; then ${\chi }_{i}v$ is the corresponding local limit. Testing against compactly supported smooth functions shows that its derivative limits are the weak derivatives of $v$. The norm equivalence proves $v_j\to v$ in $H_h^1$. The displayed norm comes from the positive-definite inner product ${\int }_M(vw+⟨dv,dw{⟩}_h)d{\mu }_h$, so $H^1(M)$ is a real Hilbert space. Its closed subspace $H_0^1(M)$, defined as the closure of $C_c^{\mathrm{\infty }}(M\setminus \mathrm{\partial }M)$ in this norm, is Hilbert: a Cauchy sequence converges in $H^1(M)$ and its limit lies in that closure.

The inclusion $H^1(M)\to L^2(M)$ is compact by applying Rellich with $p=q=2$, $n=3$, on the finitely many smooth auxiliary domains and taking finitely many successive subsequences; the extension theorem verifies the domain hypothesis. The local bounded trace operators give zero trace for every member of $H_0^1(M)$ by its defining closure. If Poincaré failed, choose $v_j\in H_0^1(M)$ with $\mathrm{\parallel }{v}_j{\mathrm{\parallel }}_2=1$ and $\mathrm{\parallel }d{v}_j{\mathrm{\parallel }}_2\to 0$. Compactness gives an $L^2$ limit $v$. In charts, distributional testing gives $dv=0$; moreover $v_j\to v$ in $H_h^1$, so $v\in H_0^1(M)$. Zero-gradient constancy in connected coordinate interiors, and agreement on overlaps, give one constant on each component of $M$; the interior of each connected boundary manifold is connected, since a path may be pushed off the boundary in finitely many half-charts. Each component has a boundary chart, whose flat open boundary piece has positive surface measure. Its constant has zero trace there, hence is zero. This contradicts $\mathrm{\parallel }v{\mathrm{\parallel }}_2=1$. Thus $\mathrm{\parallel }v{\mathrm{\parallel }}_2\le C\mathrm{\parallel }dv{\mathrm{\parallel }}_2$ on $H_0^1(M)$. Choice is used only as permitted in the analytic suppliers and the subsequence argument.

On this Hilbert space the form $a(v,w)={\int }_M⟨dv,dw{⟩}_{h}d{\mu }_h$ is bounded and coercive by Poincaré. The functional $w\mapsto -{\int }_{M}fwd{\mu }_h$ is bounded by Cauchy–Schwarz, since $f$ is continuous and the compact manifold has finite volume. Lax–Milgram gives one weak solution $u\in H_0^1(M)$, and it is unique there. In a chart its equation is ${\mathrm{\partial }}_i(A^{ij}{\mathrm{\partial }}_{j}u)=mf$, where $m=\sqrt{\det h}$ and $A^{ij}=m{h}^{ij}$. The smooth matrix $A$ is uniformly positive on every fixed compact patch. To match the positive-principal-part convention for $-{\mathrm{\partial }}_i(A^{ij}{\mathrm{\partial }}_j)$ in the weak regularity supplier, take forcing $-mf$. On overlapping aligned boundary charts the weak-equation invariance input applies to this solution with locally $L^2$ forcing and smooth uniformly elliptic coefficients. After a cutoff, its zero-Dirichlet assertion applies by the defining compactly supported smooth approximations to $u$. These solution-specific assertions are used only here; the arbitrary-function construction used step 1.1.

In interior patches apply the declared nested-domain interior regularity lemma with $k=2$; $mf\in H^2$ and all coefficient bounds hold. At a boundary, localize with nested smooth cutoffs on smooth auxiliary domains of step 2.1. The localized function has zero Dirichlet data on the flat portion by the defining smooth approximations to $u$, and vanishes near every artificial boundary. The cutoff commutator contains only smooth multiples of $u$ and $Du$. Initially it is $L^2$, so the boundary theorem with $k=0$ gives local $H^2$. On a smaller patch the new commutator is now $H^1$ and $mf\in H^1$; use $k=1$ for $H^3$. On one further smaller patch it is $H^2$ and $mf\in H^2$; use $k=2$ for $H^4$. The arbitrary-function pullback bounds of step 1.1 and the multiplier bounds of [F4] justify each localization. Thus no undeclared interior counterpart or premature differentiation is needed. Smooth auxiliary domains satisfy the Sobolev extension hypothesis, so $H^4\hookrightarrow C^{2,\gamma }$ for any fixed $0<\gamma <1\mathrm{/}2$. The weak equation is therefore classical and its zero trace is its continuous boundary value.

Write the classical coordinate equation as $Lu=h^{ij}{u}_{ij}+b^j{u}_j=f$, with smooth $b^j=m^{-1}{\mathrm{\partial }}_i(m{h}^{ij})$ and zero reaction coefficient. The interior second-order Schauder interface gives $C^{2,\alpha }$ on smaller interior patches. On a boundary auxiliary domain apply the global boundary interface to $\eta u$: its forcing is $\eta f+[L,\eta ]u$, and its entire boundary value is zero. Here $u\in C^{2,\gamma }$ already, so $[L,\eta ]u$ has bounded first derivatives on the fixed compact patch and is $C^{0,\alpha }$ for every $0<\alpha <1$. Hence this application gives $C^{2,\alpha }$ up to the boundary as well. Now the imported higher-order regularity theorem applies with $k=2$ to each classical chart solution: the coefficients are smooth, $f\in C^{2,\alpha }$, the physical boundary is smooth and the data there are zero. Its local-boundary-portion version ignores artificial chart edges. It gives $u\in C^{4,\alpha }$ on smaller patches, hence on $M$ by a finite cover. This use is of the imported regularity theorem, not an assertion that second-order estimates alone license differentiating a merely $H^4$ solution.

It remains to obtain the stated estimate with only the larger-chart data and supremum. Fix enough nested subpatches between the specified smaller and larger patches. Let $E_j(V)=\mathrm{\parallel }u{\mathrm{\parallel }}_{C^{j,\alpha }(V)}$ and $G_j(V)=\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{j,\alpha }(V)}$. The interior second-order estimate or the imported local boundary estimate gives $E_2(V^{\mathrm{\prime }})\le C(G_0(V)+\mathrm{\parallel }u{\mathrm{\parallel }}_{0;V})$. All subsequent differentiations are legitimate by step 7.1. In the interior, $L(D_{l}u)=D_{l}f-(D_l{h}^{ij})u_{ij}-(D_l{b}^j)u_j$; the second-order estimate on nested patches gives $E_3(V^{\mathrm{\prime }})\le C(G_1(V)+E_2(V))$. Differentiate twice and apply it again to obtain $E_4(V^{\mathrm{\prime }})\le C(G_2(V)+E_3(V))$. At a flattened boundary, apply the local second-order estimate first to $D_{t}u$ and then to $D_s{D}_{t}u$, with $s,t$ tangential. Their boundary values are zero, and the differentiated forcing uses coefficient derivatives through order two and solution derivatives of strictly lower order. For order three these estimates control derivatives with at least one tangential factor; the equation solved for $u_{nn}$, divided by $h^{nn}\ge \lambda >0$, recovers $u_{nnn}$. For order four the two-tangential estimates control all derivatives with at most two normal factors. Apply $D_t{D}_n$ to the same identity to recover those with three normal factors, then $D_n^2$ to recover $u_{nnnn}$; all other highest-order terms have fewer normal factors, and coefficient-derivative terms use derivatives of $u$ of order at most three. This proves the same two recurrences at the boundary. Composing them on the nested patches gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{4,\alpha ;U^{\mathrm{\prime }}}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{2,\alpha ;U}+\mathrm{\parallel }u{\mathrm{\parallel }}_{0;U})$. Constants use only the ellipticity, coefficient $C^{2,\alpha }$ bounds and fixed chart separations; in particular this argument does not require a global topology restriction on $U$.

Any classical solution with zero boundary values belongs to $H_0^1(M)$: cut it off in boundary charts at distance $\epsilon$ from the flat face. Since it is $C^1$ and vanishes there, $u=O(x_n)$, so the cutoff error and its gradient tend to zero in $L^2$ on the shrinking boundary layer; finite partitions and interior smoothing give the defining compactly supported smooth approximations. The homogeneous weak energy identity may therefore be tested with the difference of two solutions, giving ${\int }_M\mathrm{\mid }du{\mathrm{\mid }}_h^{2}d{\mu }_h=0$. Poincaré gives $u=0$. Together with the constructed solution and step 8.1, this proves the Statement.