Constant-field branch on a ball

The base fields are smooth and ${\mathrm{\partial }}_i{\bar{E}}_0^i={\mathrm{\partial }}_i{\bar{B}}_0^i=0$, so the source in [F1] is the constant $S=a^2$, and scaling them by $\epsilon$ preserves their zero divergences. The units are consistent: $a^2$ and $\mathrm{\Delta }$ have units of inverse length squared, while $\psi$ and $\epsilon$ are dimensionless. At any point of $\mathrm{\partial }{B}_R$, rotate that point to $(0,0,R)$. In a sufficiently small neighborhood its boundary is $x_3=(R^2-\mathrm{\mid }{x}^{\mathrm{\prime }}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$ and the ball occupies the side below this graph. The graph is smooth for $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }<R$, so the ball is a bounded $C^k$ domain for every finite $k$. By [A1] and [F12] it is a $W^{k,2}$-extension domain for each $k\ge 1$; all uses of [F9] below therefore meet that hypothesis.

Smoothness bootstrap, initial gain. Set $u=u_{\epsilon }={\psi }_{\epsilon }-1$ and $f=({\epsilon }^2{a}^2\mathrm{/}4){\psi }_{\epsilon }^{-3}$, so $-\mathrm{\Delta }u=f$ and $u=0$ on $\mathrm{\partial }\mathrm{\Omega }$. The branch gives $u\in C^{2,\alpha }(\bar{\mathrm{\Omega }})$ and ${\psi }_{\epsilon }\ge 1$. The identities ${\mathrm{\partial }}_i({\psi }^{-3})=-3{\psi }^{-4}{\mathrm{\partial }}_i\psi$ and ${\mathrm{\partial }}_{ij}({\psi }^{-3})=12{\psi }^{-5}({\mathrm{\partial }}_i\psi )({\mathrm{\partial }}_j\psi )-3{\psi }^{-4}{\mathrm{\partial }}_{ij}\psi$ follow from [F11] and the product rule and show that $f\in C^2(\bar{\mathrm{\Omega }})$, with bounded derivatives through order two, hence $f\in H^2(\mathrm{\Omega })$ by integration by parts against compactly supported tests. By [F6], $u\in H_0^1(\mathrm{\Omega })$. Its classical equation tested against $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ gives the weak identity; continuity of both sides in the $H^1$ test norm and the defining density of $C_c^{\mathrm{\infty }}$ in $H_0^1$ extend it to every test in $H_0^1$. Applying [F8] with $m=2$ yields $u\in H^4(\mathrm{\Omega })$.

On $B_R(0)$, define $v(x):=\frac{a^2}{24}(R^2-\mathrm{\mid }x{\mathrm{\mid }}^2)$. It vanishes on $\mathrm{\partial }{B}_R$, and $\mathrm{\Delta }\mathrm{\mid }x{\mathrm{\mid }}^2=6$ in ${\mathbb{R}}^3$, so $-8\mathrm{\Delta }v=-8(-6a^2\mathrm{/}24)=2a^2=2S$. By uniqueness of the zero-boundary solution in [F2], this is the coefficient in the branch expansion.

Smoothness bootstrap, composition and iteration. Fix an integer $m\ge 4$ and suppose $\psi =1+u\in H^m(\mathrm{\Omega })$. Its continuous range is a compact interval $J\subset [1,\mathrm{\infty })$ by [F2] and [F9]. To verify ${\psi }^{-3}\in H^m$, use [F10] to choose smooth restrictions ${\psi }_n\to \psi$ in $H^m$; [F9] gives uniform convergence, so for all sufficiently large $n$ the values of ${\psi }_n$ lie in one compact interval $J^{\mathrm{\prime }}\subset (0,\mathrm{\infty })$ containing $J$. For each smooth $q$ with values in $J^{\mathrm{\prime }}$, repeated application of the total chain and product rules [F11] expresses every $D^{\beta }(q^{-3})$, $\mathrm{\mid }\beta \mathrm{\mid }\le m$, as a finite sum of bounded factors $G^{(r)}(q)$, $G(s)=s^{-3}$, times products ${\prod }_j{D}^{{\gamma }_j}q$ with $\mathrm{\mid }{\gamma }_j\mathrm{\mid }\ge 1$ and ${\sum }_j\mathrm{\mid }{\gamma }_j\mathrm{\mid }=\mathrm{\mid }\beta \mathrm{\mid }$. Each product is in $L^2$ with a bound controlled by $\mathrm{\parallel }q{\mathrm{\parallel }}_{H^m}$: derivatives of order at most $m-2$ lie in $H^2$ or a higher Sobolev space and hence in $L^{\mathrm{\infty }}$ by [F9]; a factor of order $m$ can occur only alone and is in $L^2$; a factor of order $m-1$ occurs at most once, lies in $H^1\subset L^6$, and any remaining factor has total derivative order at most one and lies in $L^{\mathrm{\infty }}$. Thus $q\mapsto G^{\mathrm{\prime }}(q)$ is uniformly bounded in $H^m$ on the bounded smooth approximants and their line segments. The same difference identity applied to $G^{\mathrm{\prime }}$ shows $t\mapsto G^{\mathrm{\prime }}({\psi }_{\mathrm{\ell }}+t({\psi }_n-{\psi }_{\mathrm{\ell }}))$ is continuous in $H^m$, so its displayed integral exists as a Banach-valued Riemann integral. The identity $G({\psi }_n)-G({\psi }_{\mathrm{\ell }})=({\psi }_n-{\psi }_{\mathrm{\ell }}){\int }_0^1{G}^{\mathrm{\prime }}({\psi }_{\mathrm{\ell }}+t({\psi }_n-{\psi }_{\mathrm{\ell }}))dt$, together with the Banach-algebra bound [F9], shows $G({\psi }_n)$ is Cauchy in $H^m$. Its uniform limit is $G(\psi )$, proving ${\psi }^{-3}\in H^m$. Hence $f\in H^m$; [F8] gives $u\in H^{m+2}$. Starting with $m=4$ and repeating for $m=4,6,8,\dots$ proves $u\in H^{2j}(\mathrm{\Omega })$ for every sufficiently large integer $j$. The same embeddings imply $u\in H^r(\mathrm{\Omega })$ for every finite integer $r$, and then $u\in C^k(\bar{\mathrm{\Omega }})$ for every $k$, so ${\psi }_{\epsilon }\in C^{\mathrm{\infty }}(\bar{\mathrm{\Omega }})$.

Verification of the full constraints. On the oriented manifold without boundary $\mathrm{\Sigma }=\mathrm{\Omega }$, take the conformal seed data $\bar{h}=\delta$, $\bar{A}=0$, $\tau =0$, $\mathrm{\Lambda }=0$, $\bar{E}=\epsilon a{e}_1$, and $\bar{B}=0$. Smoothness and positivity of ${\psi }_{\epsilon }$ follow from steps 1.2–2.2 and [F2], so [F3] defines smooth $h,K,E,B$ and gives $K=0$. The Maxwell constraints hold because ${\mathrm{\partial }}_i{\bar{E}}^i={\mathrm{\partial }}_i{\bar{B}}^i=0$ by step 1.1 and the barred metric is Euclidean. Since $\bar{R}=0$ and $\mathrm{\mid }\bar{E}{\mathrm{\mid }}_{\delta }^2={\epsilon }^2{a}^2$, the transformed Hamiltonian constraint [F4] is exactly $-8\mathrm{\Delta }{\psi }_{\epsilon }=2{\epsilon }^2{a}^2{\psi }_{\epsilon }^{-3}$, the branch equation. Its transformed momentum equation has zero left side because $K=\tau =0$ and zero right side because $\bar{E}{\times }_{\delta }\bar{B}=0$. Thus all four constraints in [F5] hold. The data are smooth constrained time-symmetric data on this fixed ball; the Dirichlet boundary value is prescribed, and the proof makes no claim of an isolated asymptotically flat black hole or any other boundary-value problem.