Planar critical vortices for arbitrary finite divisors

Choose radii $R_k\to \mathrm{\infty }$ with every center in $D_{R_k}$. Let $s=2{\sum }_j{m}_j\log \mathrm{\mid }z-z_j\mathrm{\mid }$ and $w_k=u_{R_k}-s$. For each compact disk $K$ and each $0<\alpha <1$, [F2] gives a uniform $C^{2,\alpha }$ bound on $w_k$ on a slightly larger disk. Thus $(w_k,D{w}_k,D^2{w}_k)$ is uniformly bounded and equicontinuous there. Apply the Euclidean sequence statement [F6] to the finite derivative tuple on each nonempty compact disk. A countable diagonal selection under [A1] then gives locally uniform convergence of $w_k$ and all its derivatives through order two. The fundamental theorem of calculus on coordinate segments identifies the limiting derivative tuple as the first and second derivatives of one $C^2$ function w. Thus the subsequence converges in $C^2$ on every compact disk. Set $h=s+w$.

On every compact set, $e^{u_{R_k}}=e^s{e}^{w_k}$ converges uniformly to $e^h$. Passing to the limit in the weak divisor equation for $u_{R_k}$ gives $\mathrm{\Delta }h+\kappa (1-e^h)=4\pi D$ distributionally. Since $s-2m_j\log \mathrm{\mid }z-z_j\mathrm{\mid }$ is smooth near each center and $w$ is $C^2$ there, the prescribed logarithmic subtraction extends continuously. The equation for $w$ is $\mathrm{\Delta }w=\kappa (e^s{e}^w-1)$ with smooth coefficient $e^s=\mathrm{\mid }P{\mathrm{\mid }}^2$; repeated interior Schauder regularity [F4] bootstraps $w$ to $C^{\mathrm{\infty }}$, so each center subtraction is smooth. The bounds $u_{-}\le u_{R_k}\le 0$ pass to the limit, and $u_{-}(z)\to 0$ at infinity; hence $h\le 0$ and $h\to 0$.

If $h_1,h_2$ are two solutions with the same divisor and boundary condition, $v=h_1-h_2$ extends smoothly across the centers. Subtracting their equations gives $\mathrm{\Delta }v-c(z)v=0$, where $c(z)=\kappa {\int }_0^1{e}^{(1-t)h_2+t{h}_1}dt\ge 0$, extended continuously by $0$ at the centers. On $D_R$, the weak maximum principle [F7] applied to $v$ and $-v$ bounds $\mathrm{\mid }v\mathrm{\mid }$ by ${\sup }_{\mathrm{\partial }{D}_R}\mathrm{\mid }v\mathrm{\mid }$. This boundary supremum tends to $0$ as $R\to \mathrm{\infty }$, so $v=0$ and the scalar solution is unique. When $D$ is empty, $h=0$ is a solution and the same uniqueness argument shows it is the only one.

Suppose $N>0$ and put $w=-h\ge 0$. Away from the divisor, $\mathrm{\Delta }w=\kappa (1-e^{-w})\le \kappa w$. On $\mathrm{\Omega }=({\mathbb{R}}^2\setminus \{z_j\})\times (-\pi \mathrm{/}(2\sqrt{\kappa }),\pi \mathrm{/}(2\sqrt{\kappa }))$, define $U(z,t)=-w(z)\cos (\sqrt{\kappa }t)$. Then $U\le 0$ and ${\mathrm{\Delta }}_{z,t}U=\cos (\sqrt{\kappa }t)(\kappa w-\mathrm{\Delta }w)\ge 0$. The finite-punctured plane is path connected (polygonal paths can detour around finitely many points), hence $\mathrm{\Omega }$ is connected. If $w$ vanished at a point, $U$ would attain its maximum $0$ in $\mathrm{\Omega }$; [F8] would force $U\equiv 0$, so $w\equiv 0$ off the divisor, contradicting $w(z)\to +\mathrm{\infty }$ at each positive-multiplicity center. Thus $h<0$ away from the divisor.

Fix $0<\alpha <\sqrt{\kappa }$. Since $w=-h\to 0$, choose $c$ with ${\alpha }^2<c<\kappa$ and $R_0$ so large that $\kappa (1-e^{-w})\ge cw$ for $\mathrm{\mid }z\mathrm{\mid }\ge R_0$. Let $\phi (z)=C{e}^{-\alpha (\mathrm{\mid }z\mathrm{\mid }-R_0)}$ with $C\ge {\sup }_{\mathrm{\mid }z\mathrm{\mid }=R_0}w$. For $r=\mathrm{\mid }z\mathrm{\mid }\ge R_0$, $\mathrm{\Delta }\phi =({\alpha }^2-\alpha \mathrm{/}r)\phi <c\phi$, so $(\mathrm{\Delta }-c)(w-\phi )\ge 0$. The function $w-\phi$ is nonpositive on $\mathrm{\mid }z\mathrm{\mid }=R_0$ and tends to $0$ at infinity. If it were positive somewhere, its positive maximum would be attained at a finite interior point, where $(\mathrm{\Delta }-c)(w-\phi )<0$, a contradiction. Therefore $0\le w\le \phi$, proving the claimed scalar exponential bound.

Let $P(z)={\prod }_j(z-z_j{)}^{m_j}$ and $s=2\log \mathrm{\mid }P\mathrm{\mid }$. The factor $h-s$ is smooth on all of $\mathbb{C}$. Define $q=\sqrt{\xi }{e}^{(h-s)\mathrm{/}2}P$ and $A=-\frac{1}{2}\ast d(h-s)$; both extend smoothly over the centers and $q$ has exactly the prescribed zeros. On the punctured plane write $P=\mathrm{\mid }P\mathrm{\mid }{e}^{i\chi }$; holomorphicity gives $d\chi =\frac{1}{2}\ast ds$, hence $A=d\chi -\frac{1}{2}\ast dh$. Writing $q=\sqrt{\xi }{e}^{h\mathrm{/}2}{e}^{i\chi }$ and using $\ast dh=(-{\mathrm{\partial }}_{2}h)d{x}_1+({\mathrm{\partial }}_{1}h)d{x}_2$ gives $D_{A,1}q=\frac{1}{2}{e}^{i\chi }\sqrt{\xi }{e}^{h\mathrm{/}2}({\mathrm{\partial }}_{1}h-i{\mathrm{\partial }}_{2}h)$ and $D_{A,2}q=\frac{1}{2}{e}^{i\chi }\sqrt{\xi }{e}^{h\mathrm{/}2}({\mathrm{\partial }}_{2}h+i{\mathrm{\partial }}_{1}h)$, so $D_{A,1}q+i{D}_{A,2}q=0$.

To obtain derivative decay, fix $\alpha <{\alpha }^{\mathrm{\prime }}<\sqrt{\kappa }$ and use step 3.3 at rate ${\alpha }^{\mathrm{\prime }}$. On each unit ball centered at large $z$, both $w$ and $\mathrm{\Delta }w=\kappa (1-e^{-w})$ are $O(e^{-{\alpha }^{\mathrm{\prime }}\mathrm{\mid }z\mathrm{\mid }})$. By [F3] and [F5], $w$ has a $C^{1,\beta }$ norm of that order on the concentric half-ball for some $\beta >0$. Thus the right-hand side has a $C^{0,\beta }$ norm of the same order, and [F4] gives $C^{2,\beta }$ decay on a smaller ball. Inductively, for each multi-index $\gamma$, differentiating $\mathrm{\Delta }w=\kappa (1-e^{-w})$ gives $\mathrm{\Delta }{D}^{\gamma }w=\kappa e^{-w}{D}^{\gamma }w+G_{\gamma }$, where $G_{\gamma }$ is a finite sum of products of lower derivatives of $w$ and is exponentially small by the induction hypothesis. Applying [F3] and [F4] on nested unit balls gives the same $e^{-{\alpha }^{\mathrm{\prime }}\mathrm{\mid }z\mathrm{\mid }}$ bound for two more derivatives. Since $e^{-{\alpha }^{\mathrm{\prime }}\mathrm{\mid }z\mathrm{\mid }}\le e^{-\alpha \mathrm{\mid }z\mathrm{\mid }}$ outside a fixed disk, every finite derivative has the stated rate.

The curvature is $dA=-\frac{1}{2}\mathrm{\Delta }(h-s)d{x}_1\wedge d{x}_2$. Since $\mathrm{\Delta }s=4\pi D$ and $h$ satisfies the divisor equation, $\mathrm{\Delta }(h-s)=-\kappa (1-e^h)$; therefore $B=\frac{\kappa }{2}(1-e^h)=e^2(\xi -\mathrm{\mid }q{\mathrm{\mid }}^2)$. On the exterior of a disk, $e^{-i\chi }$ is a smooth local gauge, not asserted to extend across the zeros; it gives $q_{\mathrm{e}\mathrm{x}\mathrm{t}}=\sqrt{\xi }{e}^{h\mathrm{/}2}$ and $A_{\mathrm{e}\mathrm{x}\mathrm{t}}=-\frac{1}{2}\ast dh$. The phase satisfies $\chi =N\theta +O(r^{-1})$ because $P(z)=z^N{\prod }_j(1-z_j\mathrm{/}z{)}^{m_j}$. Thus the gauge-invariant $B,D_{A}q$ decay exponentially by step 4.1, while no exponential decay of the raw phase correction is claimed.

The reconstructed fields are smooth on compact sets. In the exterior gauge of step 5.1, $B$, $D_{A}q$, and $\mathrm{\mid }q{\mathrm{\mid }}^2-\xi =\xi (e^h-1)$ all decay exponentially, so each squared term in the defining energy is integrable on ${\mathbb{R}}^2$; hence the energy is finite. To compute flux, write $v=h-s$, so $\mathrm{\Delta }v=-\kappa (1-e^h)$. For $r>0$ the chain rule in polar coordinates gives $r\mathrm{\Delta }v={\mathrm{\partial }}_r(r{\mathrm{\partial }}_{r}v)+r^{-1}{\mathrm{\partial }}_{\theta \theta }v$. Smoothness of $v$ on ${\bar{D}}_R$ bounds both ${\mathrm{\partial }}_r(r{\mathrm{\partial }}_{r}v)$ and $r^{-1}{\mathrm{\partial }}_{\theta \theta }v$ uniformly for $0<r\le R$: by the chain rule each is a sum of bounded first derivatives and $r$ times bounded second derivatives. Extending them arbitrarily at $r=0$ makes them Borel and absolutely integrable on the finite Lebesgue product $[0,R]\times [0,2\pi ]$. Fubini [F11] therefore permits either integration order. The angular derivative integrates to zero over $0\le \theta \le 2\pi$ by periodicity. Apply the polar formula [F9] to the positive and negative parts of ${\mathbf{1}}_{D_R}\mathrm{\Delta }v$; using [F11] and $ds=rd\theta$ then gives ${\int }_{D_R}\mathrm{\Delta }vd{x}_{1}d{x}_2={\int }_0^{2\pi }{\int }_0^R{\mathrm{\partial }}_r(r{\mathrm{\partial }}_{r}v)drd\theta ={\int }_0^{2\pi }R{\mathrm{\partial }}_{r}v(R,\theta )d\theta ,$ since $v$ is smooth and $r{\mathrm{\partial }}_{r}v\to 0$ at $r=0$. By step 4.1, ${\mathrm{\partial }}_{r}h$ decays exponentially; from $s=2N\log r+O(r^{-1})$, ${\mathrm{\partial }}_{r}s=2N\mathrm{/}R+O(R^{-2})$ uniformly in $\theta$. Therefore the boundary integral tends to $-4\pi N$. By [F10], $\kappa {\int }_{{\mathbb{R}}^2}(1-e^h)=4\pi N$; using $B=\frac{\kappa }{2}(1-e^h)$ gives flux $2\pi N$. For the empty divisor $h=0$, the reconstruction is the flat constant vacuum and has zero flux.