Bounded-domain charged conformal problem

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a nonempty, bounded, connected open set with smooth boundary, meaning each boundary point has Euclidean coordinates in which $\mathrm{\Omega }$ is one side of a smooth graph. Use Euclidean coordinates $x=(x^1,x^2,x^3)$, let $\delta ={\sum }_{i=1}^{3}d{x}^i\otimes d{x}^i$, write ${\mathrm{\partial }}_i=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^i$, and set $\mathrm{\Delta }={\sum }_{i=1}^3{\mathrm{\partial }}_i^2$. A multi-index is $\beta =({\beta }_1,{\beta }_2,{\beta }_3)\in {\mathbb{N}}_0^3$, where ${\mathbb{N}}_0=\{0,1,2,\dots \}$; write $\mathrm{\mid }\beta \mathrm{\mid }={\beta }_1+{\beta }_2+{\beta }_3$ and $D^{\beta }={\mathrm{\partial }}_1^{{\beta }_1}{\mathrm{\partial }}_2^{{\beta }_2}{\mathrm{\partial }}_3^{{\beta }_3}$. Fix $0<\alpha <1$, a real function $S\in C^{0,\alpha }(\bar{\mathrm{\Omega }})$ with $S\ge 0$, and a real dimensionless parameter $\epsilon$. For a real function $g$ on $\bar{\mathrm{\Omega }}$, define

$[g{]}_{0,\alpha }:={\sup }_{\begin{array}{c}x,y\in \bar{\mathrm{\Omega }}\\ x\mathrm{\ne }y\end{array}}\frac{\mathrm{\mid }g(x)-g(y)\mathrm{\mid }}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }},\mathrm{\parallel }g{\mathrm{\parallel }}_{C^{0,\alpha }}:={\sup }_{\bar{\mathrm{\Omega }}}\mathrm{\mid }g\mathrm{\mid }+[g{]}_{0,\alpha },$

and let $C^{0,\alpha }(\bar{\mathrm{\Omega }})$ consist of the continuous functions with finite displayed norm. Define $C^{2,\alpha }(\bar{\mathrm{\Omega }})$ to consist of functions whose derivatives through order $2$ on $\mathrm{\Omega }$ extend continuously to $\bar{\mathrm{\Omega }}$ and whose norm

$\mathrm{\parallel }u{\mathrm{\parallel }}_{C^{2,\alpha }}:={\sum }_{\mathrm{\mid }\beta \mathrm{\mid }\le 2}{\sup }_{\bar{\mathrm{\Omega }}}\mathrm{\mid }{D}^{\beta }u\mathrm{\mid }+{\sum }_{\mathrm{\mid }\beta \mathrm{\mid }=2}[D^{\beta }u{]}_{0,\alpha }$

is finite, where $\beta$ is a multi-index. Write $X:=\{u\in C^{2,\alpha }(\bar{\mathrm{\Omega }}):u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0\}$, with its inherited norm, and $Y:=C^{0,\alpha }(\bar{\mathrm{\Omega }})$. The bounded-domain charged conformal problem is to find $u\in X$ such that $\psi :=1+u>0$ on $\bar{\mathrm{\Omega }}$ and

$-8\mathrm{\Delta }\psi =2{\epsilon }^{2}S{\psi }^{-3}\text{in }\mathrm{\Omega },\psi {\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=1.$

In the conformal-seed specialization $\bar{h}=\delta$, $\bar{A}=0$, $\tau =0$, and $\mathrm{\Lambda }=0$ (so $h={\psi }^4\delta$ and $K=0$), the initial data are time-symmetric, meaning their second fundamental form is $K=0$. The scalar equation is the Hamiltonian constraint from the conformal electrovacuum transformation when $S=\mathrm{\mid }{\bar{E}}_0{\mathrm{\mid }}_{\delta }^2+\mathrm{\mid }{\bar{B}}_0{\mathrm{\mid }}_{\delta }^2$. Here ${\bar{E}}_0,{\bar{B}}_0:\bar{\mathrm{\Omega }}\to {\mathbb{R}}^3$ are independently prescribed contravariant Euclidean vector fields, each the restriction of a smooth map on an open neighborhood of $\bar{\mathrm{\Omega }}$, and divergence-free means ${\sum }_{i=1}^3{\mathrm{\partial }}_i{\bar{E}}_0^i={\sum }_{i=1}^3{\mathrm{\partial }}_i{\bar{B}}_0^i=0$ on $\mathrm{\Omega }$ in these Cartesian coordinates. The actual barred seed fields are $\bar{E}=\epsilon {\bar{E}}_0$ and $\bar{B}=\epsilon {\bar{B}}_0$. In geometric length coordinates, $\psi ,\epsilon$ are dimensionless, ${\bar{E}}_0,{\bar{B}}_0$ have units of inverse length, and $S$ has units of inverse length squared. This is a specialization with prescribed fields; the general problem allows any nonnegative $S\in C^{0,\alpha }(\bar{\mathrm{\Omega }})$. The definition does not assert existence, uniqueness, a momentum constraint, or an asymptotically Euclidean or puncture solution.

Conformal electrovacuum seed data

Exact conformal electrovacuum constraint transformation