Disclinations in limiting Landau-de Gennes theory

TL;DR

This paper investigates the low-temperature behavior of Landau-de Gennes theory, revealing multiple energy minimizers with distinct structures, including biaxial torus and split-core segment configurations.

Contribution

It identifies multiple minimizers in the low-temperature limit of Landau-de Gennes theory, highlighting structural differences in the solutions.

Findings

Existence of at least two distinct minimizers

One minimizer exhibits biaxial torus structure

Another minimizer has split-core segment structure

Abstract

In this article we study the low-temperature limit of a Landau-de Gennes theory. Within all S2-valued R-axially symmetric maps (see Definition 1.1), the limiting energy functional has at least two distinct energy minimizers. One minimizer has biaxial torus structure, while another minimizer has split-core segment structure on the z-axis.

Equations

$$\displaystyle\int_{B_{R}}\dfrac{1}{2}\hskip 1.0pt\big{|}\hskip 1.0pt\nabla\mathscr{Q}\hskip 1.0pt\big{|}^{2}-\dfrac{a^{2}}{2}\hskip 1.0pt\big{|}\hskip 1.0pt\mathscr{Q}\hskip 1.0pt\big{|}^{2}-\sqrt{6}\hskip 2.0pt\mathrm{tr}\big{(}\mathscr{Q}^{3}\big{)}+\dfrac{1}{2}\hskip 1.0pt\big{|}\hskip 1.0pt\mathscr{Q}\hskip 1.0pt\big{|}^{4}.$$

$$-\Delta \mathcal{Q}=a^2\mathcal{Q}+3\sqrt{6}\left({\mathcal{Q}}^2-\frac{1}{3}|\mathcal{Q}{|}^2{\mathbf{I}}_3\right)-2|\mathcal{Q}{|}^2\mathcal{Q}\text{in BR.}\text{vspace}0.2pc$$

$$\displaystyle e_{\rho}=\left(\frac{x_{1}}{\rho},\frac{x_{2}}{\rho},0\right),\hskip 20.0pte_{\theta}=\left(-\frac{x_{2}}{\rho},\frac{x_{1}}{\rho},0\right),\hskip 20.0pte_{z}=\big{(}0,0,1\big{)}.$$

$$\mathcal{Q}=\sum _{j=1}^3{q}_j{M}_j,\text{where for j=1,2,3, qj=qj(ρ,z) are real-valued functions.}$$

$$\displaystyle a^{-1}\mathscr{Q}\left(Rx\right)=\overline{\mathscr{Q}}\left[u\right]:=u_{1}\left(e_{\rho}\otimes e_{\rho}-\dfrac{1}{2}\hskip 0.5pt\mathbf{I}_{2}\right)+\dfrac{\sqrt{3}}{2}\hskip 1.0ptu_{2}\hskip 1.0ptM_{2}+\dfrac{1}{2}\hskip 0.5ptu_{3}\hskip 1.0ptM_{3},\hskip 10.0pt\text{where $\mathbf{I}_{2}=\left(\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&0\\ \end{array}\right),$}$$

$$\displaystyle\left\{\begin{array}[]{lcl}-\Delta u_{1}+\dfrac{4}{\rho^{2}}u_{1}\hskip 2.0pt=a^{2}R^{2}u_{1}+\dfrac{3}{\sqrt{2}}\hskip 0.5ptaR^{2}\left(-2\hskip 0.5ptu_{1}u_{2}+\dfrac{\sqrt{3}}{2}u_{3}^{2}\right)-a^{2}R^{2}\hskip 0.5pt|\hskip 0.5ptu\hskip 0.5pt|^{2}\hskip 0.5ptu_{1},&&\text{in $\mathrm{B}_{1}$;}\vspace{0.5pc}\\ -\Delta u_{2}\hskip 33.0pt=a^{2}R^{2}u_{2}+\dfrac{3}{\sqrt{2}}\hskip 0.5ptaR^{2}\left(-u_{1}^{2}+u_{2}^{2}+\dfrac{1}{2}\hskip 0.5ptu_{3}^{2}\right)\hskip 4.0pt-a^{2}R^{2}\hskip 0.5pt|\hskip 0.5ptu\hskip 0.5pt|^{2}\hskip 0.5ptu_{2},&&\text{in $\mathrm{B}_{1}$;}\vspace{0.5pc}\\ -\Delta u_{3}+\dfrac{1}{\rho^{2}}u_{3}\hskip 2.0pt=a^{2}R^{2}u_{3}+\dfrac{3}{\sqrt{2}}\hskip 0.5ptaR^{2}\hskip 0.5pt\bigg{(}\sqrt{3}\hskip 1.0ptu_{1}u_{3}+u_{2}u_{3}\bigg{)}-a^{2}R^{2}\hskip 0.5pt|\hskip 0.5ptu\hskip 0.5pt|^{2}\hskip 0.5ptu_{3},&&\text{in $\mathrm{B}_{1}$.}\vspace{0.5pc}\\ \end{array}\right.$$

$$\displaystyle E_{a,\hskip 0.5pt\mu}\left[\hskip 0.5ptv\hskip 0.5pt\right]:=\int_{B_{1}}\big{|}\hskip 0.5pt\nabla v\hskip 1.0pt\big{|}^{2}+\frac{1}{\rho^{2}}\left(4v_{1}^{2}+v_{3}^{2}\hskip 0.5pt\right)+\dfrac{\mu}{a}\hskip 1.0ptF_{a}(\hskip 0.5ptv\hskip 0.5pt).$$

$$F_a(v):=D_a-3\sqrt{2}aP\left(v\right)+\frac{a^2}{2}{\left[|v{|}^2-1\right]}^2\text{with P(v):=−v2v12+23v1v32+31v23+21v2v32.}$$

$$u=\frac{\sqrt{2}{H}_{+}}{a}{U}^{\ast }\text{on ∂B1, where H+:=43+9+8a2andU∗:=(23r2ρ2,23(r2z2−31),3r2ρz).}$$

$$\displaystyle E_{\mu}\left[\hskip 0.5ptu\hskip 0.5pt\right]:=E\left[\hskip 0.5ptu\hskip 0.5pt\right]+\sqrt{2}\hskip 1.0pt\mu\int_{B_{1}}1-3P\left(u\right):=\int_{B_{1}}\big{|}\hskip 0.5pt\nabla u\hskip 1.5pt\big{|}^{2}+\dfrac{1}{\rho^{2}}\left(4u_{1}^{2}+u_{3}^{2}\hskip 1.5pt\right)+\sqrt{2}\hskip 1.0pt\mu\hskip 1.5pt\big{(}\hskip 0.5pt1-3\hskip 1.0ptP\big{(}\hskip 0.5ptu\hskip 0.5pt\big{)}\hskip 0.5pt\big{)}$$

$$\displaystyle\mathscr{F}:=\Big{\{}v=v(\rho,z)\in\mathbb{S}^{2}:E_{\mu}[\hskip 1.0ptv\hskip 1.0pt]<\infty\hskip 3.0pt\text{and}\hskip 3.0pt\text{$v=U^{*}$ on ${\partial}B_{1}$}\Big{\}}.$$

$$\displaystyle L\left[\hskip 1.0ptu\hskip 1.0pt\right]:=\big{(}\hskip 1.0ptu_{1}\cos 2\theta,\hskip 2.0ptu_{1}\sin 2\theta,\hskip 2.0ptu_{2},\hskip 2.0ptu_{3}\cos\theta,\hskip 2.0ptu_{3}\sin\theta\hskip 1.0pt\big{)}.$$

$$\displaystyle-\dfrac{1}{2}\left(u_{1}+\dfrac{1}{\sqrt{3}}\hskip 0.5ptu_{2}\right);\hskip 5.0pt\dfrac{1}{4}\bigg{\{}\left(u_{1}+\dfrac{1}{\sqrt{3}}\hskip 1.0ptu_{2}\right)-\sqrt{\big{(}u_{1}-\sqrt{3}\hskip 0.5ptu_{2}\big{)}^{2}+4u_{3}^{2}}\hskip 1.0pt\bigg{\}};\hskip 5.0pt\dfrac{1}{4}\bigg{\{}\left(u_{1}+\dfrac{1}{\sqrt{3}}\hskip 1.0ptu_{2}\right)+\sqrt{\big{(}u_{1}-\sqrt{3}\hskip 0.5ptu_{2}\big{)}^{2}+4u_{3}^{2}}\hskip 1.0pt\bigg{\}}.$$

$$\displaystyle\mathrm{Hess}_{U^{*}}\big{[}\hskip 0.5pt\mathrm{P}_{f},\mathrm{P}_{f}\hskip 0.5pt\big{]}=\dfrac{32\hskip 1.0pt\pi}{5}\left[\hskip 1.5pt\int_{0}^{1}\big{(}f^{\prime}\big{)}^{2}r^{2}-3\int_{0}^{1}f^{2}\hskip 1.0pt\right]+\dfrac{72\hskip 1.0pt\sqrt{2}}{5}\hskip 1.0pt\pi\hskip 1.0pt\mu\hskip 1.0pt\int_{0}^{1}f^{2}\hskip 1.0ptr^{2}.$$

$$\displaystyle\mathrm{Hess}_{\hskip 1.0ptU^{*}}\big{[}\hskip 1.0pt\mathrm{P}_{g},\mathrm{P}_{g}\hskip 0.5pt\big{]}<0.$$

$$\displaystyle\mathrm{Min}\hskip 2.0pt\bigg{\{}\hskip 1.0ptE_{\mu}\left[\hskip 1.0ptu\right]:u\in\mathscr{F}_{\hskip 1.0ptb,+}\bigg{\}}\hskip 2.0pt<\hskip 2.0ptE_{\mu}\hskip 1.0pt\big{[}\hskip 1.0ptU^{*}\hskip 1.0pt\big{]},\hskip 30.0pt\mathrm{Min}\hskip 2.0pt\bigg{\{}\hskip 1.0ptE_{\mu}\left[\hskip 1.0ptu\right]:u\in\mathscr{F}_{\hskip 1.0ptc,-}\hskip 1.0pt\bigg{\}}\hskip 2.0pt<\hskip 2.0ptE_{\mu}\hskip 1.0pt\big{[}\hskip 1.0ptU^{*}\hskip 1.0pt\big{]},$$

$$\displaystyle\left\{\begin{array}[]{lcl}-\Delta w-\dfrac{3\sqrt{2}}{2}\hskip 1.0pt\mu\hskip 1.0pt\nabla_{w}S=\left\{\hskip 1.0pt\hskip 1.0pt\big{|}\hskip 1.0pt\nabla w\hskip 1.0pt\big{|}^{2}-\dfrac{9\sqrt{2}}{2}\hskip 1.0pt\mu\hskip 1.0ptS\left[w\right]\hskip 1.0pt\right\}w&&\text{in $B_{1}$;}\vspace{0.3pc}\\ \hskip 78.0ptw=L\left[U^{*}\right]&&\text{on ${\partial}B_{1}$.}\end{array}\right.$$

$$S\left[w\right]:=-w_3\left(w_1^2+w_2^2\right)+\sqrt{3}{w}_2{w}_4{w}_5+\frac{1}{2}{w}_3\left(w_4^2+w_5^2\right)+\frac{1}{3}{w}_3^3+\frac{\sqrt{3}}{2}{w}_1\left(w_4^2-w_5^2\right).$$

$$\Lambda^{+}:=\big{(}\hskip 0.5pt0,\hskip 2.0pt0,\hskip 2.0pt\cos\varphi,\hskip 2.0pt\sin\varphi\cos\theta,\hskip 2.0pt\sin\varphi\sin\theta\hskip 0.5pt\big{)}$$

$$\kappa \left[u\right]:=\frac{\sqrt{2}}{2}\left\{1+\frac{u_1-\sqrt{3}{u}_2}{\sqrt{{\left(u_1-\sqrt{3}{u}_2\right)}^2+4u_3^2}}\right\}{e}_{\rho }+\frac{\sqrt{2}{u}_3}{\sqrt{{\left(u_1-\sqrt{3}{u}_2\right)}^2+4u_3^2}}{e}_z.$$

$$\displaystyle\left\{\begin{array}[]{lcl}-e_{z},&&\text{if $\varphi^{\star}=-\pi$;}\vspace{0.5pc}\\ \dfrac{\sqrt{2}}{2}\left\{1-\dfrac{\varkappa\hskip 1.0pt\mathrm{ctan}\hskip 0.5pt\varphi^{\star}}{\sqrt{4+\varkappa^{2}\hskip 1.0pt\mathrm{ctan}^{2}\varphi^{\star}}}\right\}^{1/2}e_{\rho}-\sqrt{\dfrac{2}{4+\varkappa^{2}\hskip 1.0pt\mathrm{ctan}^{2}\varphi^{\star}}}\left\{1-\dfrac{\varkappa\hskip 1.0pt\mathrm{ctan}\hskip 0.5pt\varphi^{\star}}{\sqrt{4+\varkappa^{2}\hskip 1.0pt\mathrm{ctan}^{2}\varphi^{\star}}}\right\}^{-1/2}e_{z},&&\text{if $\varphi^{\star}\in(-\pi,0)$;}\vspace{0.5pc}\\ e_{\rho},&&\text{if $\varphi^{\star}=0$;}\vspace{0.5pc}\\ \dfrac{\sqrt{2}}{2}\left\{1+\dfrac{\varkappa\hskip 1.0pt\mathrm{ctan}\hskip 0.5pt\varphi^{\star}}{\sqrt{4+\varkappa^{2}\hskip 1.0pt\mathrm{ctan}^{2}\varphi^{\star}}}\right\}^{1/2}e_{\rho}+\sqrt{\dfrac{2}{4+\varkappa^{2}\hskip 1.0pt\mathrm{ctan}^{2}\varphi^{\star}}}\left\{1+\dfrac{\varkappa\hskip 1.0pt\mathrm{ctan}\hskip 0.5pt\varphi^{\star}}{\sqrt{4+\varkappa^{2}\hskip 1.0pt\mathrm{ctan}^{2}\varphi^{\star}}}\right\}^{-1/2}e_{z},&&\text{if $\varphi^{\star}\in(0,\pi)$;}\vspace{0.5pc}\\ e_{z},&&\text{if $\varphi^{\star}=\pi$.}\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{lcl}e_{z},&&\text{if $\varphi_{+}^{\star}=0$;}\vspace{0.5pc}\\ \dfrac{\sqrt{2}}{2}\left\{1-\dfrac{\sqrt{3}\hskip 1.0pt\cos\varphi_{+}^{\star}}{\sqrt{3+\sin^{2}\varphi_{+}^{\star}}}\right\}^{1/2}e_{\rho}+\sqrt{\dfrac{2\hskip 1.0pt\sin^{2}\varphi_{+}^{\star}}{3+\sin^{2}\varphi_{+}^{\star}}}\left\{1-\dfrac{\sqrt{3}\hskip 1.0pt\cos\varphi_{+}^{\star}}{\sqrt{3+\sin^{2}\varphi_{+}^{\star}}}\right\}^{-1/2}e_{z},&&\text{if $\varphi_{+}^{\star}\in\big{(}0,\pi\big{]}$.}\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{lcl}\dfrac{\sqrt{2}}{2}\left\{1+\dfrac{\sqrt{3}\hskip 1.0pt\cos\varphi_{-}^{\star}}{\sqrt{3+\sin^{2}\varphi_{-}^{\star}}}\right\}^{1/2}e_{\rho}-\sqrt{\dfrac{2\hskip 1.0pt\sin^{2}\varphi_{-}^{\star}}{3+\sin^{2}\varphi_{-}^{\star}}}\left\{1+\dfrac{\sqrt{3}\hskip 1.0pt\cos\varphi_{-}^{\star}}{\sqrt{3+\sin^{2}\varphi_{-}^{\star}}}\right\}^{-1/2}e_{z},&&\text{if $\varphi_{-}^{\star}\in\big{[}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}$;}\vspace{0.5pc}\\ -e_{z},&&\text{if $\varphi_{-}^{\star}=\pi$.}\end{array}\right.$$

$$\displaystyle\lim_{b\hskip 1.0pt\rightarrow\hskip 1.0pt-1}E_{\mu}\left[u_{b}^{+}\right]=E_{\mu}\left[u_{b_{\star}}^{+}\right]=\mathrm{Min}\hskip 2.0pt\bigg{\{}\hskip 1.0ptE_{\mu}\left[\hskip 1.0ptu\right]:u\in\mathscr{F}^{s}\bigg{\}},\hskip 20.0pt\lim_{c\hskip 1.0pt\rightarrow\hskip 1.0pt1}E_{\mu}\left[u_{c}^{-}\right]=E_{\mu}\left[u_{c_{\star}}^{-}\right]=\mathrm{Min}\hskip 2.0pt\bigg{\{}\hskip 1.0ptE_{\mu}\left[\hskip 1.0ptu\right]:u\in\mathscr{F}^{s}\bigg{\}}.$$

$$\displaystyle E_{\mu}\left[\hskip 0.5ptu\hskip 0.5pt\right]=2\pi E_{\hskip 1.0pt\mathbb{D}}\left[\hskip 0.5ptu\hskip 0.5pt\right]:=2\pi\int_{\mathbb{D}}\left\{\hskip 1.5pt\big{|}\hskip 1.0pt{\partial}_{\rho}u\hskip 1.0pt\big{|}^{2}+\big{|}\hskip 1.0pt{\partial}_{z}u\hskip 1.0pt\big{|}^{2}+\dfrac{1}{\rho^{2}}\big{(}\hskip 1.0pt4u_{1}^{2}+u_{3}^{2}\hskip 1.0pt\big{)}+\sqrt{2}\hskip 1.0pt\mu\hskip 1.0pt\big{(}\hskip 1.0pt1-3\hskip 1.0ptP\hskip 0.5pt\big{(}\hskip 1.0ptu\hskip 1.0pt\big{)}\hskip 1.0pt\big{)}\hskip 1.5pt\right\}\rho\hskip 2.0pt\mathrm{d}\rho\hskip 2.0pt\mathrm{d}z.$$

$$\displaystyle-\dfrac{1}{\rho}\hskip 1.0ptD\cdot\big{(}\rho\hskip 1.0ptDu\big{)}+\hskip 2.0pt\dfrac{1}{\rho^{2}}\hskip 1.0pt\left(\begin{array}[]{lcl}4u_{1}\\ \hskip 4.0pt0\\ \hskip 3.0ptu_{3}\end{array}\right)-\dfrac{3\sqrt{2}}{2}\hskip 1.0pt\mu\hskip 1.0pt\nabla_{u}P=\left\{\hskip 2.0pt\hskip 1.0pt\big{|}\hskip 1.0ptDu\hskip 1.0pt\big{|}^{2}+\dfrac{1}{\rho^{2}}\left(4u_{1}^{2}+u_{3}^{2}\right)-\dfrac{9\sqrt{2}}{2}\hskip 1.0pt\mu\hskip 1.0ptP\hskip 0.5pt\big{(}\hskip 1.0ptu\hskip 1.0pt\big{)}\hskip 2.0pt\right\}u\hskip 15.0pt\text{in $\mathbb{D}$.}$$

$$u_{b;2}^{+}\ge b\text{and}{u}_{c;2}^{-}\le c\text{on T, respectively.}$$

$$\displaystyle\mathscr{I}_{\sigma}:=\bigg{\{}\left(x,r\right)\in D^{+}_{\sigma}\left(x_{*}\right)\times\mathbb{R}_{+}:D_{r}\left(x\right)\subset D_{\sigma}^{+}\left(x_{*}\right)\hskip 1.0pt\bigg{\}}\hskip 3.0pt\bigcup\hskip 3.0pt\bigg{\{}\left(x,r\right)\in T\times\mathbb{R}_{+}:D_{r}^{+}\left(x\right)\subset D_{\sigma}^{+}\left(x_{*}\right)\hskip 1.0pt\bigg{\}}.$$

$$\displaystyle E_{x,\hskip 1.0ptr}^{+}\left[\hskip 0.5ptu\hskip 0.5pt\right]:=\int_{D_{r}^{+}\left(x\right)}\left\{\hskip 1.5pt\big{|}\hskip 1.0ptDu\hskip 1.0pt\big{|}^{2}+\dfrac{1}{\rho^{2}}\big{(}\hskip 1.0pt4u_{1}^{2}+u_{3}^{2}\hskip 1.0pt\big{)}+\sqrt{2}\hskip 1.0pt\mu\hskip 1.0pt\big{(}\hskip 1.0pt1-3\hskip 1.0ptP\big{(}\hskip 1.0ptu\hskip 1.0pt\big{)}\hskip 1.0pt\big{)}\hskip 1.5pt\right\}\rho\hskip 2.0pt\mathrm{d}\rho\hskip 1.0pt\mathrm{d}z.$$

$$(1).E_{x,{\theta }_{1}r}^{+}\left[u_b^{+}\right]\le r^{3/2};(2).E_{x,{\theta }_{1}r}^{+}\left[u_b^{+}\right]\le \frac{1}{2}{E}_{x,r}^{+}\left[u_b^{+}\right].$$

Full text

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A $5$-vector field on $B_{1}$ is called $\mathscr{R}$-axially symmetric if it equals to $L\left[u\right]$ for some $\mathscr{R}$-axially symmetric $3$-vector field $u$ on $B_{1}$. Moreover we denote by $\mathscr{F}^{s}$ the subspace of $\mathscr{F}$ which contains all $\mathscr{R}$-axially symmetric maps in $\mathscr{F}$.

Given a $\mathscr{R}$-axially symmetric vector, we use the following definition to describe its regularity at a point:

Eigenvalues and eigenvectors of the order parameter $\mathscr{Q}$ play key roles in the study of Landau-de Gennes theory. In light of our ansatz (1.4), the eigenvalues of $\mathscr{Q}$ can be represented in terms of the variable $u$. Therefore we introduce

[TABLE]

These three eigenvalues are denoted by $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, respectively and will be referred as eigenvalues determined by $u$ in the following. At a location $x\in B_{1}$, we call

1. (1).

$u$* is isotropic at $x$ if $x$ is a point singularity of $u$ (see Definition 1.2);* 2. (2).

$u$* is uniaxial at $x$ if two of the three eigenvalues determined by $u$ are identical and different from zero at $x$;* 3. (3).

$u$* is biaxial at $x$ if all the three eigenvalues determined by $u$ are different at $x$.*

Moreover a vector field is called director field determined by $u$ if its values are normalized eigenvectors corresponding to the largest eigenvalue in $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$.

With the above definitions, we can define the so-called biaxial torus and split-core line segment.

I.3. MAIN RESULTS AND STRUCTURE OF THE ARTICLE

For any $\mu\geq 0$, $U^{*}$ defined in (1.7) is a critical point of $E_{\mu}$. However it is not an $E_{\mu}$-energy minimizer within the configuration space $\mathscr{F}$ (see (1.9)). For any smooth radial function $f$ with $f(1)=0$, we let $\mathrm{P}_{f}$ be the orthogonal projection of \big{(}\hskip 1.0pt0,\hskip 0.5ptf,\hskip 0.5pt0\hskip 1.0pt\big{)} to the tangent plane of $\mathbb{S}^{2}$ at $U^{*}$. The Hessian of $E_{\mu}$ at $U^{*}$ then satisfies

[TABLE]

In light of Lemma 1.3 in [58], there is a smooth radial function $g$ compactly supported on $(0,1)$ so that

[TABLE]

Therefore $U^{*}$ is linearly unstable in $\mathscr{F}$. There exists at least one map in $\mathscr{F}^{s}$ (see Definition 1.1) whose $E_{\mu}$-energy is strictly less than the $E_{\mu}$-energy of $U^{*}$. We are now concerned about energy-minimizers of $E_{\mu}$ in $\mathscr{F}^{s}$.

Our strategy to prove Theorem 1.5 relies on a novel bifurcation argument. In the following $B_{1}^{+}$ ($B_{1}^{-}$ resp.) denotes the open upper-half (lower-half resp.) part of $B_{1}$. $T$ is the flat boundary of $B_{1}^{+}$. Moreover we let \mathrm{I}_{-}:=\big{(}-1,\hskip 1.0pt-1\hskip 0.5pt/\hskip 0.5pt2\hskip 1.5pt\big{]} and \mathrm{I}_{+}:=\big{[}-1\hskip 0.5pt/\hskip 0.5pt2,\hskip 1.0pt1\hskip 1.5pt\big{)}. For any $b\in\mathrm{I}_{-}$ and $c\in\mathrm{I}_{+}$, we define \mathscr{F}_{\hskip 1.0ptb,+}:=\Big{\{}u\in\mathscr{F}^{s}:u_{2}\hskip 2.0pt\geq\hskip 2.0ptb\hskip 6.0pt\text{\mathrm{on}\hskip 3.0ptT}\hskip 1.0pt\Big{\}} and \mathscr{F}_{\hskip 1.0ptc,-}:=\Big{\{}\hskip 1.0ptu\in\mathscr{F}^{s}:u_{2}\hskip 2.0pt\leq\hskip 2.0ptc\hskip 6.0pt\text{\mathrm{on}\hskip 3.0ptT}\hskip 1.0pt\Big{\}}. Note that for $g$ satisfying (1.11), $|\hskip 1.0ptg\hskip 1.0pt|$ also satisfies (1.11). For real number $\sigma$ with sufficiently small absolute value, the normalized map of $U^{*}+\sigma\hskip 1.0pt\mathrm{P}_{|\hskip 0.5ptg\hskip 0.5pt|}$, denoted by $U_{\sigma}^{*}$, has strictly less $E_{\mu}$-energy than $U^{*}$. Since $U_{\sigma}^{*}\hskip 1.0pt\in\hskip 1.0pt\mathscr{F}_{\hskip 1.0ptb,+}$ if $\sigma\geq 0$ and $U_{\sigma}^{*}\in\mathscr{F}_{\hskip 1.0ptc,-}$ if $\sigma\leq 0$, it then follows

[TABLE]

for any $b\in\mathrm{I}_{-}$ and $c\in\mathrm{I}_{+}$. By direct method of calculus of variations, for any $b\in\mathrm{I}_{-}$, the first minimization problem in (1.12) can be solved by a minimizer $u_{b}^{+}$. In light of $P$ defined in (1.6), we have P\big{(}\hskip 0.5ptv_{1},v_{2},v_{3}\hskip 0.5pt\big{)}\hskip 2.0pt\leq\hskip 2.0ptP\big{(}\hskip 0.5pt|\hskip 0.5ptv_{1}\hskip 0.5pt|,v_{2},|\hskip 0.5ptv_{3}\hskip 0.5pt|\hskip 0.5pt\big{)} for any $(v_{1},v_{2},v_{3})\in\mathbb{R}^{3}$. Hence we can make the first and third components of $u_{b}^{+}$ non-negative on $B_{1}^{+}$. Similarly for any $c\in\mathrm{I}_{+}$, we can find a minimizer, denoted by $u_{\hskip 1.0ptc}^{-}$, of the second minimization problem in (1.12). Moreover the first and third components of $u_{\hskip 1.0ptc}^{-}$ are non-negative throughout $B_{1}^{+}$. By (1.12), $u_{b}^{+}$ and $u_{c}^{-}$ solve some Signorini-type problems for $\mathbb{S}^{2}$-valued mappings (see [12] and [53] for scalar case).

Regularity of the two minimizers (i.e. $u_{b}^{+}$ and $u_{c}^{-}$) are crucial while we study biaxial torus and split-core segment structures. By the augment operator $L$, we reduce the regularity problem for $u_{b}^{+}$ and $u_{c}^{-}$ to the associated regularity problem of L\hskip 1.0pt\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]} and L\hskip 1.0pt\big{[}\hskip 1.0ptu_{c}^{-}\hskip 1.0pt\big{]}, respectively. In the remaining L\hskip 1.0pt\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]} and L\hskip 1.0pt\big{[}\hskip 1.0ptu_{c}^{-}\hskip 1.0pt\big{]} are referred as bifurcation solutions. To study regularity of these two bifurcation solutions, we split $\overline{B_{1}}$ into several sub-regions. By Hélen’s trick and arguments of Rivi$\grave{\text{e}}$re-Struwe in [54] (see also [49]), the bifurcation solutions are smooth in $B_{1}\sim\left(l_{3}\cup T\right)$. Here $l_{3}$ denotes the $z$-axis. Moreover by [49], these two bifurcation solutions are continuous up to the points on ${\partial}B_{1}\sim\left(l_{3}\cup T\right)$. As for the south and north poles, we can flatten ${\partial}B_{1}$ near these two poles and extend symmetrically the definition of these bifurcation solutions in the new coordinates from a half ball to a whole ball. By Rivi$\grave{\text{e}}$re-Struwe arguments in [54] and Schoen-Unlenbeck’s geometric Lemma 2.5 in [57], these two bifurcation solutions are continuous up to these two poles. Hence we only need to study regularity of bifurcation solutions on $B_{1}\cap l_{3}$ and the free boundary $T$. In Sect.II, we consider a Signorini-type problem for harmonic maps with potential. We show that the bifurcation solutions are smooth on T^{\circ}:=T\sim\Big{\{}(0,0),(1,0)\Big{\}}. Without ambiguity we still use $T$ to denote the set \Big{\{}\left(\rho,0\right):\rho\in\big{[}\hskip 0.5pt0,1\hskip 0.5pt\big{]}\hskip 1.0pt\Big{\}}. By this regularity result, the bifurcation solutions solve weakly the following Dirichlet boundary value problem:

[TABLE]

Here $S\hskip 0.5pt[\cdot]$ is defined by

[TABLE]

Results in [49] therefore infer the continuity of the bifurcation solutions up to ${\partial}B_{1}\cap T$. In addition an $\epsilon$-regularity result, i.e. Proposition 3.1, also holds for the bifurcation solutions at the origin. In Sect.III, we show that the bifurcation solutions are indeed smooth at the origin. Hence if the bifurcation solutions have singularities, then singularities must be on $l_{3}$ and different from the origin and two poles. Sect.IV is devoted to studying asymptotic behaviour of bifurcation solutions near their singularities on $l_{3}$. With the preparations in Sections II-IV, in Sect.V, we study in details the biaxial torus and split-core segment structures of these bifurcation solutions. Our second main result is about specific properties of L\hskip 1.0pt\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]}, particularly its biaxial torus structure.

1. (1).

The augmented map L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptb}^{+}\big{]} is a weak solution to the Dirichlet boundary value problem (1.13). It is smooth on $B_{1}$, except possibly a finite number of singularities on $l_{3}$. L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptb}^{+}\big{]} is smooth at the origin, south pole and north pole. If L\hskip 1.0pt\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]} has singularities, then it must have even number of singularities on $l_{\hskip 1.0pt3}^{+}:=B_{1}^{+}\cap l_{3}$. By symmetry, L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptb}^{+}\big{]} has the same number of singularities on $l_{3}^{-}:=B_{1}^{-}\cap l_{3}$ as the number of singularities that it has on $l_{3}^{+}$; 2. (2).

If L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptb}^{+}\hskip 0.5pt\big{]} has singularities, then at each singularity on $l_{\hskip 1.0pt3}^{+}$, tangent map of L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptb}^{+}\hskip 0.5pt\big{]} must equal to

[TABLE]

The tangent map of L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptb}^{+}\hskip 0.5pt\big{]} at its lowest singularity on $l_{\hskip 1.0pt3}^{+}$ is given by $\Lambda^{-}$ in (1.14). Moreover on $l_{3}^{+}$, tangent maps of two consecutive singularities of L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptb}^{+}\hskip 0.5pt\big{]} must be different maps in (1.14); 3. (3).

In the sense of Definition 1.2, $u_{\hskip 1.0ptb}^{+}$ is regular away from singularities of L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptb}^{+}\hskip 0.5pt\big{]}. The singularities of L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptb}^{+}\hskip 0.5pt\big{]}, if it has, must also be singularities of $u_{\hskip 1.0ptb}^{+}$. If we understand $u_{\hskip 1.0ptb}^{+}$ as a map defined on the $\rho$ - $z$ plane, then $u_{\hskip 1.0ptb}^{+}$ is continuous on $T$ and analytic on its interior part $T^{\circ}$. $u_{\hskip 1.0ptb}^{+}$ takes the value \left(\sqrt{3}\big{/}2,1\big{/}2,0\right) at finitely many points on $T^{\circ}$. At these points, $u_{\hskip 1.0ptb}^{+}$ is uniaxial; 4. (4).

There is a $\rho_{0}\in(0,1)$ so that for an $\epsilon>0$ sufficiently small, the second component of $u_{\hskip 1.0ptb}^{+}$ satisfies u_{\hskip 1.0ptb;\hskip 1.0pt2}^{+}>1\big{/}2 on \Big{\{}\left(\rho,0\right):\rho\in\left(\rho_{0}-\epsilon,\rho_{0}\right)\Big{\}} and u_{\hskip 1.0ptb;\hskip 1.0pt2}^{+}<1\big{/}2 on \Big{\{}\left(\rho,0\right):\rho\in\left(\rho_{0},\rho_{0}+\epsilon\right)\Big{\}}. Here and in what follows $u_{\hskip 1.0ptb;\hskip 0.5ptj}^{+}$ denotes the $j$-th component of $u_{\hskip 1.0ptb}^{+}$. Suppose that $D_{\epsilon}\left(\hskip 0.5pt\rho_{0},0\hskip 0.5pt\right)$ is the disk in the $\rho$ - $z$ plane with center \big{(}\rho_{0},0\big{)} and radius $\epsilon$. Then $u_{\hskip 1.0ptb}^{+}$ is biaxial on D_{\epsilon}\left(\hskip 0.5pt\rho_{0},0\hskip 0.5pt\right)\sim\Big{\{}\big{(}\rho_{0},0\big{)}\Big{\}}, provided $\epsilon$ is sufficiently small. More precisely the three eigenvalues in Definition 1.3 computed in terms of $u_{\hskip 1.0ptb}^{+}$ satisfy the quantitative relationship $\lambda_{3}\hskip 2.0pt>\hskip 2.0pt\lambda_{2}\hskip 2.0pt>\hskip 2.0pt\lambda_{1}$ on D_{\epsilon}\big{(}\rho_{0},0\big{)}\sim\Big{\{}\big{(}\rho_{0},0\big{)}\Big{\}}; 5. (5).

For any given $3$-vector field $u=(u_{1},u_{2},u_{3})$, it determines a vector field

[TABLE]

Then on D_{\epsilon}\left(\hskip 0.5pt\rho_{0},0\hskip 0.5pt\right)\sim\Big{\{}\big{(}\rho_{0},0\big{)}\Big{\}}, the normalized vector field of \kappa\hskip 1.0pt\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]}, denoted by $\kappa^{\star}$ for simplicity, is a director field determined by $u_{b}^{+}$ (see Definition 1.3). It is the normalized eigenvector corresponding to $\lambda_{3}$ in Definition 1.3 computed in terms of $u_{\hskip 1.0ptb}^{+}$. $\kappa^{\star}$ is continuous on D_{\epsilon}\left(\hskip 0.5pt\rho_{0},0\hskip 0.5pt\right)\sim\Big{\{}\big{(}\rho,0\big{)}:\rho\in\big{(}\hskip 0.5pt\rho_{0}-\epsilon,\rho_{0}\hskip 0.5pt\big{]}\hskip 1.0pt\Big{\}}. However it is not continuous up to the branch cut \Big{\{}\big{(}\rho,0\big{)}:\rho\in\big{(}\hskip 0.5pt\rho_{0}-\epsilon,\rho_{0}\hskip 0.5pt\big{]}\hskip 1.0pt\Big{\}}. Suppose that $\epsilon^{\prime}$ is an arbitrary number in $(0,\epsilon)$ and denote by $x^{\prime}$ the point \big{(}\rho_{0}-\epsilon^{\prime},0\big{)}. $\kappa^{\star}$ converges to $-e_{z}$ if we approach $x^{\prime}$ from the lower-half part of ${\partial}D_{\epsilon^{\prime}}\left(\rho_{0},0\right)$. Meanwhile $\kappa^{\star}$ converges to $e_{z}$ if we approach $x^{\prime}$ from the upper-half part of ${\partial}D_{\epsilon^{\prime}}\left(\rho_{0},0\right)$. During the process when we rotate counter-clockwisely from $x^{\prime}$ to itself along ${\partial}D_{\epsilon^{\prime}}\left(\rho_{0},0\right)$, the image of $\kappa^{\star}$ varies continuously from $-e_{z}$ to $e_{z}$ and keeps on the right-half part of $\rho$ - $z$ plane for all points on {\partial}D_{\epsilon^{\prime}}\left(\rho_{0},0\right)\sim\big{\{}x^{\prime}\big{\}}. The angle of $\kappa^{\star}$ is changed by $\pi$ during this process; 6. (6).

Let $\varphi^{\star}$ be an angular variable ranging from \big{[}-\pi,\pi\hskip 0.5pt\big{]}. The value of $u_{\hskip 1.0ptb}^{+}\left(\left(\rho_{0},0\right)+\epsilon^{\prime}\left(\cos\varphi^{\star},\sin\varphi^{\star}\right)\right)$ at $-\pi$ ($\pi$* resp.) is taken to be the limit of $u_{\hskip 1.0ptb}^{+}$ at $x^{\prime}$ along the lower (upper resp.) part of ${\partial}D_{\epsilon^{\prime}}\left(\rho_{0},0\right)$. Then the tangent map of $\kappa^{\star}$, i.e. the limit of $\kappa^{\star}\left(\left(\rho_{0},0\right)+\epsilon^{\prime}\left(\cos\varphi^{\star},\sin\varphi^{\star}\right)\right)$ as $\epsilon^{\prime}\rightarrow 0^{+}$, equals to*

[TABLE]

Here $\varkappa$ is a non-negative constant given by \varkappa=\left.\dfrac{{\partial}_{\rho}\hskip 0.5ptu_{\hskip 1.0ptb;1}^{+}-\sqrt{3}\hskip 1.5pt{\partial}_{\rho}\hskip 0.5ptu_{\hskip 1.0ptb;\hskip 0.5pt2}^{+}}{{\partial}_{z}\hskip 0.5ptu_{\hskip 1.0ptb;3}^{+}}\hskip 1.0pt\right|_{\big{(}\rho_{0},0\big{)}}. Note that by Hopf’s lemma, ${\partial}_{z}\hskip 0.5ptu_{\hskip 1.0ptb;\hskip 0.5pt3}^{+}$ is strictly positive at \big{(}\rho_{0},0\big{)}.

As for $L\left[u_{c}^{-}\right]$, it exhibits split-core segment structure, which we show in the following theorem:

1. (1).

The augmented map L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptc}^{-}\hskip 0.5pt\big{]} is a weak solution to the boundary value problem (1.13). It is smooth on $B_{1}$, except a finite number of singularities on $l_{3}$. L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptc}^{-}\hskip 0.5pt\big{]} is smooth at the origin, south pole and north pole. It must have odd number of singularities on $l_{\hskip 1.0pt3}^{+}$. By symmetry, L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptc}^{-}\big{]} has the same number of singularities on $l_{3}^{-}$ as the number of singularities that it has on $l_{3}^{+}$; 2. (2).

Part (2) of Theorem 1.6 still holds for L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptc}^{-}\hskip 0.5pt\big{]}, except that the tangent map of L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptc}^{-}\hskip 0.5pt\big{]} at its lowest singularity on $l_{\hskip 1.0pt3}^{+}$ is given by $\Lambda^{+}$ in (1.14); 3. (3).

In ths sense of Definition 1.2, $u_{\hskip 1.0ptc}^{-}$ is regular away from the singularities of L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptc}^{-}\hskip 0.5pt\big{]}. Singularities of L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptc}^{-}\hskip 0.5pt\big{]} must also be singularities of $u_{\hskip 1.0ptc}^{-}$. Moreover on $\overline{B_{1}}\cap l_{3}$, $u_{\hskip 1.0ptc}^{-}$ is uniaxial except at the singularities. Suppose that $x_{0}^{+}$ is the lowest singularity on $l_{\hskip 1.0pt3}^{+}$. $x_{0}^{-}$ is the symmetric point of $x_{0}^{+}$ with respect to the $x_{1}$-$x_{2}$* plane. For two sufficiently small positive constants $\epsilon$ and $\epsilon_{1}$, we denote by $\mathrm{D}_{\epsilon,\epsilon_{1}}$ the dumbbell on $x_{1}$-$z$ plane introduced in Definition 5.1. Then $\mathrm{D}_{\epsilon,\epsilon_{1}}$ contains properly the segment connecting $x_{0}^{+}$ and $x_{0}^{-}$. Moreover $u_{\hskip 1.0ptc}^{-}$ is biaxial on $\mathrm{D}_{\epsilon,\epsilon_{1}}\sim l_{3}$. Quantitatively the three eigenvalues in Definition 1.3 computed in terms of $u_{\hskip 1.0ptc}^{-}$ satisfy $\lambda_{3}\hskip 1.0pt>\hskip 1.0pt\lambda_{1}\hskip 1.0pt>\hskip 1.0pt\lambda_{2}$ on $\mathrm{D}_{\epsilon,\epsilon_{1}}\sim l_{3}$;* 4. (4).

The normalized vector field of \kappa\hskip 1.0pt\big{[}\hskip 1.0ptu_{c}^{-}\hskip 1.0pt\big{]}, which we denote by $\kappa_{\star}$, is a director field determined by $u_{c}^{-}$ on $\mathrm{D}_{\epsilon,\epsilon_{1}}\sim l_{3}$. Suppose that $\mathscr{C}$ is the boundary of $\mathrm{D}_{\epsilon,\epsilon_{1}}$ in the $x_{1}$-$z$* plane. $\mathscr{C}^{+}$ is the part of $\mathscr{C}$ with non-negative $x_{1}$-coordinate. From the lowest point on $\mathscr{C}$ to the most top point on $\mathscr{C}$ along $\mathscr{C}^{+}$, the image of $\kappa_{\star}$ varies continuously from $-e_{z}$ to $e_{\rho}$ and then to $e_{z}$. Meanwhile the image of $\kappa_{\star}$ keeps on the right-half part of the $(\rho,z)$ plane for all points on $\mathscr{C}^{+}\sim l_{3}$;* 5. (5).

In limiting scenario, the tangent map of $\kappa_{\star}$ at $x_{0}^{+}$ is given by

[TABLE]

Here $\varphi_{+}^{\star}$ is the polar angle of the spherical coordinates with respect to the center $x_{0}^{+}$. Along the segment connecting $x_{0}^{+}$ and $x_{0}^{-}$ (not include the two end-points), $\kappa_{\star}$ equivalently equals to $e_{\rho}$. The tangent map of $\kappa_{\star}$ at $x_{0}^{-}$ is given by

[TABLE]

Here $\varphi_{-}^{\star}$ is the polar angle of the spherical coordinates with respect to the center $x_{0}^{-}$.

In the next step we prove existence result in Theorem 1.5. Note that for any $b\in\mathrm{I}_{-}$, the energy E_{\mu}\hskip 1.0pt\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]} is non-decreasing with respect to $b$. For any $c\in\mathrm{I}_{+}$, the energy E_{\mu}\hskip 1.0pt\big{[}\hskip 1.0ptu_{c}^{-}\hskip 1.0pt\big{]} is non-increasing with respect to $c$. Hence we can use a limiting process to drop the Signorini contraint from the two minimization problems in (1.12). In Sect.VI.1, we show

[TABLE]

Sect.VI.2 is devoted to proving the convergence result in Theorem 1.5.

In this article we focus on the energy functional $E_{\mu}$ of the limiting Landau-de Gennes theory. In our forthcoming paper, we will consider the biaxial and split-core structure of solutions to the original Landau-de Gennes system (1.5).

II. REGULARITY ON THE INTERIOR OF FREE BOUNDARY

Throughout the section \mathbb{D}:=\Big{\{}\hskip 0.5pt(\rho,z):z\in(-1,1),\hskip 3.0pt\rho\in\left(0,\sqrt{1-z^{2}}\hskip 1.0pt\right)\Big{\}} and \mathbb{D}^{+}:=\mathbb{D}\cap\Big{\{}\hskip 0.5pt(\rho,z):z>0\hskip 1.0pt\Big{\}}. The gradient operator on the $\rho$ - $z$ plane is denoted by D=\big{(}{\partial}_{\rho},{\partial}_{z}\big{)}. We let $D_{r}\left(x\right)$ represent the open disk in the $\left(\rho,z\right)$-plane with center $x$ and radius $r$. $D_{r}^{+}\left(x\right)$ is the subset of $D_{r}\left(x\right)$ above the $\rho$ - axis. Under the $\mathscr{R}$-axial symmetry, we can rewrite the energy functional $E_{\mu}$ in (1.8) as follows:

[TABLE]

The Euler-Lagrange equation of $E_{\hskip 1.0pt\mathbb{D}}$ can be read as

[TABLE]

Note that the second components of $u_{\hskip 1.0ptb}^{+}$ and $u_{\hskip 1.0ptc}^{-}$ satisfy the obstacle conditions:

[TABLE]

By the $\mathscr{R}$-axial symmetry and (2.2), the images of $u_{\hskip 1.0ptb}^{+}$ and $u_{\hskip 1.0ptc}^{-}$ on $T$ are supported on two circular arcs. Moreover these two arcs are properly contained in the equator of $\mathbb{S}^{2}$. Different from [30], the supporting manifold in our problem has non-empty boundary. Our problem is a Signorini-type free boundary problem for $\mathbb{S}^{2}$-valued mappings. The main result of this section is

Due to (2.2), $u_{\hskip 1.0ptb}^{+}$ and $u_{\hskip 1.0ptc}^{-}$ do not a-priorily solve weakly the equation (2.1). Standard regularity theory for harmonic maps can not be applied directly. To tackle this difficulty, we need

[TABLE]

Moreover for any $D^{+}_{r}\left(x\right)\subset\mathbb{D}^{+}$, we let

[TABLE]

There exist two constants $\theta_{1}\in\left(0,1/2\right)$ and $\epsilon>0$ so that if $\sigma\in\left(\hskip 0.5pt0,\sigma_{*}\hskip 0.5pt\right)$ and satisfies E^{+}_{x_{*},\hskip 1.0pt\sigma}\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptb}^{+}\hskip 0.5pt\big{]}<\epsilon, then for any $\left(x,r\right)\in\mathscr{I}_{\sigma}$, either one or the other of the followings holds

[TABLE]

Here $\theta_{1}$ is an universal constant. The same result holds for $u_{\hskip 1.0ptc}^{-}$.

Firstly we prove Proposition 2.1 with Lemma 2.2.

[TABLE]

Moreover we denote by $D^{+}_{*}$ the upper-half disk $D_{\hskip 1.0pt\theta_{1}r_{*}}^{+}\left(x_{*}\right)$, where $\theta_{1}$ is given in Lemma 2.2. Now we fix an arbitrary $x\in\mkern 1.5mu\overline{\mkern-1.5muD^{+}_{*}\mkern-1.5mu}\mkern 1.5mu$. It can be shown that

[TABLE]

Suppose $x\in T$. For any r\in\big{(}\hskip 1.0pt0,8\hskip 0.5ptr_{*}\theta_{1}\big{]} fixed, we let $k$ be the natural number so that

[TABLE]

Taking $\sigma=16\hskip 1.0ptr_{*}$ in Lemma 2.2, by (2.4)-(2.5), we can apply (2.3) to get

[TABLE]

If the second inequality in (2.7) holds for any $l=1,...,k$, then by (2.6), it follows

[TABLE]

Otherwise we take $l_{*}$ to be the largest number in \Big{\{}1,...,k\Big{\}} so that the first inequality in (2.7) holds. Then

[TABLE]

The above arguments infer

[TABLE]

Suppose $x\in\mkern 1.5mu\overline{\mkern-1.5muD_{*}^{+}\mkern-1.5mu}\mkern 1.5mu\sim T$ and r\in\big{(}\hskip 1.0pt0,4\hskip 1.0ptr_{*}\theta_{1}\big{]}. If $T\cap D_{r}\left(x\right)\neq\text{\O}$, then

[TABLE]

By (2.8)-(2.9), it holds $E_{x,\hskip 1.0ptr}^{+}\left[\hskip 0.5ptu\hskip 0.5pt\right]\hskip 2.0pt\leq\hskip 2.0ptE_{x^{\prime},\hskip 0.5pt2\hskip 0.5ptr}^{+}\left[\hskip 0.5ptu\hskip 0.5pt\right]\hskip 2.0pt\lesssim_{r_{*}}\hskip 2.0ptr^{\alpha}$, which infers

[TABLE]

Suppose $x\in\overline{D_{*}^{+}}\sim T$, r\in\big{(}\hskip 1.0pt0,2\hskip 1.0ptr_{*}\theta_{1}\big{]} and $T\cap D_{r}\left(x\right)=\text{\O}$. Let $j$ be the natural number so that

[TABLE]

We assume $j\geq 2$. If $j=1$, then by (2.11) and the second inequality in (2.4), it holds E_{x,\hskip 0.5ptr}^{+}\left[\hskip 0.5ptu\hskip 0.5pt\right]\hskip 2.0pt\leq\hskip 2.0pt\big{(}2\hskip 0.5ptr_{*}\hskip 0.5pt\theta_{1}^{2}\big{)}^{-1}r. Denote by $j_{*}$ the largest number in \big{\{}2,...,j\hskip 1.0pt\big{\}} so that $T\hskip 0.5pt\cap\hskip 0.5ptD_{2\hskip 1.0ptr_{*}\theta_{1}^{j_{*}-1}}\left(x\right)\neq\text{\O}$. If $j_{*}=j$, then (2.10)-(2.11) induce

[TABLE]

Now we assume $j_{*}<j$. For any natural number $l$ satisfying $l\geq j_{*}$, we can apply Lemma 2.2 to get

[TABLE]

If for any $l=j_{*},...,j-1$, the second inequality in (2.12) holds, then by (2.11), it turns out

[TABLE]

Applying (2.10) to the right-hand side above and utilizing (2.11), we obtain

[TABLE]

If the first inequality in (2.12) are satisfied by some $l$ in \Big{\{}j_{*},...,j-1\hskip 1.0pt\Big{\}}, then we take $l_{*}$ to be the largest number in \Big{\{}j_{*},...,j-1\hskip 1.0pt\Big{\}} so that the first inequality in (2.12) holds. It then follows

[TABLE]

In summary we have

[TABLE]

(2.8), (2.10) and (2.13) infer that u\in\mathrm{C}^{0,\alpha/2}\big{(}\mkern 1.5mu\overline{\mkern-1.5muD_{*}^{+}\mkern-1.5mu}\mkern 1.5mu\big{)}. Let $u_{j}$ be the $j$-th component of $u$. If $u_{2}(x_{*})=0$, then there is a $r_{1}>0$ sufficiently small so that $D_{r_{1}}\left(x_{*}\right)\subset\mathbb{D}$. Meanwhile it holds

[TABLE]

Notice that if $u=u_{\hskip 1.0ptc}^{-}$ and $c<0$, then the second component of $u_{\hskip 1.0ptc}^{-}$ cannot equal to [math] at $x_{*}$. By (2.14) and the fact that $u_{b}^{+}$ and $u_{c}^{-}$ are minimizers of the two minimization problems in (1.12), respectively, it can be shown that $u$ solves (2.1) weakly on $D_{r_{1}}\left(x_{*}\right)$. Standard elliptic regularity then yields the smoothness of $u$ on $D_{r_{1}}\left(x_{*}\right)$. If $u_{2}\left(x_{*}\right)\neq 0$, then there is a $r_{2}>0$ small enough so that $D_{r_{2}}\left(x_{*}\right)\subset\mathbb{D}$. Meanwhile \big{|}\hskip 1.0ptu_{2}\hskip 1.0pt\big{|}\hskip 2.0pt>\hskip 2.0pt0 on $D_{r_{2}}\left(x_{*}\right)$. Letting $\eta$ be an arbitrary test function compactly supported on $D_{r_{2}}\left(x_{*}\right)$, we have, by $Du_{2}=-\dfrac{u_{1}}{u_{2}}\hskip 1.0ptDu_{1}-\dfrac{u_{3}}{u_{2}}\hskip 1.0ptDu_{3}$ on $D_{r_{2}}\left(x_{*}\right)$, that

[TABLE]

Since $u$ solves the equations of $u_{1}$ and $u_{3}$ in (2.1) weakly on $D_{r_{2}}\left(x_{*}\right)$, then by the above equality, $u$ also solves the equation of $u_{2}$ in (2.1) weakly on $D_{r_{2}}\left(x_{*}\right)$. This induces the smoothness of $u$ on $D_{r_{2}}\left(x_{*}\right)$. ∎

The remaining of this section is devoted to proving Lemma 2.2. We only consider the case when $x\in T$. The case when $x$ is off the free boundary $T$ can be similarly proved. The proof relies on a blow-up process of Luckhaus type (see [43]-[44]). As in the proof of Proposition 2.1, we still use $u$ to denote either $u_{\hskip 1.0ptb}^{+}$ or $u_{\hskip 1.0ptc}^{-}$.

Suppose that we can find two sequences \Big{\{}\hskip 0.5ptx_{n}=\left(\rho_{n},0\right)\hskip 0.5pt\Big{\}}\subset T^{\circ} and \Big{\{}r_{n}\Big{\}}\subset\mathbb{R}^{+} so that the followings hold for any $n$:

[TABLE]

Here $\sigma_{*}$ is given in Lemma 2.2. Meanwhile the $E_{\hskip 1.0pt\mathbb{D}}$ - energy of $u$ on $D^{+}_{r_{n}}(x_{n})$ satisfies

[TABLE]

This limit induces

[TABLE]

Otherwise we can extract a subsequence, still denoted by \big{\{}n\big{\}}, so that for some $r_{0}>0$ and $x_{0}\in T^{\circ}$, $r_{n}\rightarrow r_{0}$ and $x_{n}\rightarrow x_{0}$ as $n\rightarrow\infty$. By (2.16), it follows $E_{x_{0},\hskip 1.0ptr_{0}}^{+}\left[\hskip 0.5ptu\hskip 0.5pt\right]=0$ which gives $u_{1}\equiv 0$ on $D_{r_{0}}^{+}(x_{0})$. Utilizing this result, the non-negativity of $u_{1}$ on $\mathbb{D}^{+}$ and the equation satisfied by $u_{1}$ (see (2.1)), we can apply the strong maximum principle to $u_{1}$ and obtain $u_{1}\equiv 0$ on $\mathbb{D}^{+}$. This result violates the boundary condition of $u$. Letting

[TABLE]

we also have

[TABLE]

[TABLE]

Using (2.16) and (2.20), we obtain

[TABLE]

The above estimate and (2.17) induce (1) in (2.19).

Utilizing (2.20) and (2) in (2.15), we can find a positive constant $c_{\hskip 0.5pt\mu,\rho_{*}}$ depending on $\mu$ and $\rho_{*}$ so that

[TABLE]

By (2.17) and the above estimate, we get \displaystyle\int_{D_{r_{n}}^{+}\left(x_{n}\right)}\big{|}\hskip 1.0ptD\hskip 0.5ptu\hskip 1.0pt\big{|}^{2}\hskip 2.0pt>\hskip 2.0pt\dfrac{r_{n}^{3/2}}{2}, provided $n$ is large. Recall the definition of $\epsilon_{n}^{2}$ in (2.18). The last lower bound infers

[TABLE]

(2) in (2.19) then follows by this upper bound and (2.17). ∎

Now we rescale domain and value of $u$ by letting

[TABLE]

Here we still use $D_{r}\left(x\right)$ to denote the disk in the $\xi$-space with center $x$ and radius $r$. Moreover $D_{r}^{+}\left(x\right)$ is still the upper part of $D_{r}\left(x\right)$ where the second coordinate $\xi_{2}>0$. For simplicity if the center is [math], then we use $D_{r}$ and $D_{r}^{+}$ to represent $D_{r}(0)$ and $D_{r}^{+}(0)$, respectively. With aid of Poincaré’s inequality and the definition of $\epsilon_{n}$ in (2.18), $u_{n}$ is uniformly bounded in $H^{1}(D_{1}^{+})$. Then there exists a $u_{\infty}\in H^{1}(D^{+}_{1})$ so that up to a subsequence,

[TABLE]

Moreover for any $p\in\left(1,2\right)$ we also have uniform boundedness of $u_{n}$ in $W^{1,p}\left(D_{1}^{+}\right)$. Therefore by Theorem 6.2 in [50], we can keep extracting a subsequence, still denoted by \big{\{}u_{n}\big{\}}, so that

[TABLE]

The limit $u_{\infty}$ is harmonic in $D_{1}^{+}$. That is

The proof of Lemma 2.4 is similar to proof of (3.13) in [21]. We omit it here. From Lemma 2.4, we can not imply the regularity of $u_{\infty}$ up to the flat boundary T_{1}:=\Big{\{}\left(\hskip 0.5pt\xi_{1},0\hskip 0.5pt\right):\xi_{1}\in\left(-1,1\right)\Big{\}}. To achieve the boundary regularity of $u_{\infty}$, we need to figure out the range of $u_{\infty}$ on $D_{1}^{+}$ and $T_{1}$. Before proceeding, let us introduce some notations. We use $\Gamma$ to denote the supporting manifold of $u$ on $T$. More precisely $\Gamma$ equals to either

[TABLE]

Let $U\left(\Gamma\right)\subset\mathbb{R}^{3}$ be a small neighborhood of $\Gamma$ in $\mathbb{R}^{3}$. For any $x\in U\left(\Gamma\right)$, we denote by $\Pi_{\Gamma}\left(x\right)$ the unique point on $\Gamma$ so that \big{|}\hskip 1.0ptx-\Pi_{\Gamma}\left(x\right)\hskip 1.0pt\big{|}=\min\Big{\{}\hskip 1.0pt|\hskip 1.0ptx-y\hskip 1.0pt|:y\in\Gamma\hskip 1.0pt\Big{\}}. The operator $\Pi_{\Gamma}$ is Lipschitz continuous from $U\left(\Gamma\right)$ to $\Gamma$. In the remaining arguments, we need to keep extracting subsequences. Without mentioning, for a sequence in one of lemmas below, it should satisfy its associated limits obtained before that lemma.

Now we begin to consider the range of $u_{\infty}$ on $D_{1}^{+}$ and $T_{1}$. Firstly we have

[TABLE]

By trace theorem on Lipschitz domain and Poincaré inequality, the above estimate can be reduced to \mathrm{d}^{2}\big{(}y_{n},\Gamma\big{)}\hskip 2.0pt\lesssim\hskip 2.0pt\epsilon_{n}^{2}. Here we also have used (2.18). The proof then follows. ∎

By this lemma, up to a subsequence, still denoted by $n$, we have

[TABLE]

The next lemma concerns about the range of $u_{\infty}$ when it is restricted on $D_{1}^{+}$.

[TABLE]

Here $\mathrm{Tan}_{y_{*}}\hskip 1.0pt\mathbb{S}^{2}$ is the $2$-plane in $\mathbb{R}^{3}$ which contains all vectors perpendicular to $y_{*}$.

[TABLE]

In light of (2.25), the second term on the right-hand side of (2.26) converges to $Y_{*}$. By (2) in (2.22), $u_{n}$ converges to $u_{\infty}$ for almost all $\xi\in D_{1}^{+}$. Therefore the limit of

[TABLE]

exists for almost all $\xi\in D_{1}^{+}$. Since $\Pi_{\Gamma}\left(y_{n}\right)\in\Gamma$, up to a subsequence, there is a $y_{*}\in\Gamma$ so that $\Pi_{\Gamma}\left(y_{n}\right)\rightarrow y_{*}$ as $n\rightarrow\infty$. Then for almost all $\xi\in D_{1}^{+}$, $u\left(x_{n}+r_{n}\hskip 1.0pt\xi\right)$ converges to $y_{*}$ also as $n\rightarrow\infty$. Here we have used (1) in (2.19). Hence the limit of the ratio in (2.27) is perpendicular to $y_{*}$ for almost all $\xi\in D_{1}^{+}$. This infers $u_{\infty}\in Y_{*}+\mathrm{Tan}_{y_{*}}\mathbb{S}^{2}$ for almost all $\xi\in D_{1}^{+}$. The proof follows since by Lemma 2.4, $u_{\infty}$ is smooth in $D_{1}^{+}$. ∎

As for the boundary values of $u_{\infty}$ on $T_{1}$, we need to consider the following three cases:

[TABLE]

For cases I and II, we have

[TABLE]

Here $\mathrm{Tan}_{y_{*}}\hskip 1.0pt\Gamma$ is the $1$-dimensional line in $\mathbb{R}^{3}$ which contains all vectors perpendicular to $y_{*}$ and \big{(}0,0,1\big{)}.

In fact (2.29) still holds for III in (2.28). But in this case we can have more detailed result. That is

[TABLE]

Here for $y_{*}\in{\partial}\hskip 2.0pt\Gamma$, we use $\mathrm{Tan}^{+}_{y_{*}}\hskip 1.0pt\Gamma$ to denote the half line in $\mathrm{Tan}_{y_{*}}\hskip 1.0pt\Gamma$ so that for any $v\in\mathrm{Tan}_{y_{*}}^{+}\hskip 1.0pt\Gamma$, it holds

[TABLE]

Similarly $\mathrm{Tan}^{-}_{y_{*}}\hskip 1.0pt\Gamma$ denotes the half line in $\mathrm{Tan}_{y_{*}}\hskip 1.0pt\Gamma$ so that for any $v\hskip 1.0pt\in\hskip 1.0pt\mathrm{Tan}^{-}_{y_{*}}\hskip 1.0pt\Gamma$, it holds $-v\hskip 1.0pt\in\hskip 1.0pt\mathrm{Tan}^{+}_{y_{*}}\hskip 1.0pt\Gamma$.

[TABLE]

Here we have used $\Pi_{\Gamma}\left(y_{n}\right)\in\Gamma$ and $\Pi_{\Gamma}\left(y_{n}\right)\rightarrow y_{*}$ as $n\rightarrow\infty$. By (2.26), we can further decompose $u_{n}$ into

[TABLE]

In light of (2.23), (2.25), (2.31), as $n\rightarrow\infty$, the limit of the last term in (2.32) exists for $\mathscr{H}^{1}$-almost all $\xi\in T_{1}$. Hence for $\mathscr{H}^{1}$-almost all $\xi\in T_{1}$, $u\left(x_{n}+r_{n}\hskip 0.5pt\xi\right)\in\Gamma$ and converges to $y_{*}$ as $n\rightarrow\infty$. This infers

[TABLE]

Using this result, (2.23), (2.25) and (2.31), we can take $n\rightarrow\infty$ in (2.32) to get (2.30). ∎

Motivated by Lemmas 2.6-2.8, we introduce the following two configurations spaces:

[TABLE]

and

[TABLE]

Here $\sigma$ is an arbitrary number in $\left(\hskip 0.5pt0,1\hskip 0.5pt\right]$. $T_{\sigma}$ denotes the set \Big{\{}\left(\hskip 1.0pt\xi_{1},0\hskip 1.0pt\right):\xi_{1}\in\left(-\sigma,\sigma\right)\hskip 1.0pt\Big{\}}. We claim that if I or II (III resp.) in (2.28) holds, then $u_{\infty}$ is an energy minimizing map in $\mathscr{F}_{A,\hskip 1.0pt1}$ ($\mathscr{F}_{B,\hskip 1.0pt1}$ resp.). See the definition of energy minimizing map in Definition 2.10. This claim will be shown in Lemma 2.11. In the following we consider the uniform convergence of $u_{n}$ on $S_{\sigma}^{+}$, where for any $\sigma\in\left(0,1\right)$, S_{\sigma}^{+}:=\Big{\{}\hskip 1.0pt\xi=\left(\hskip 1.0pt\xi_{1},\xi_{2}\hskip 1.0pt\right):|\hskip 1.0pt\xi\hskip 1.0pt|=\sigma\hskip 4.0pt\text{and}\hskip 4.0pt\xi_{2}\geq 0\hskip 1.0pt\Big{\}}.

1. (1).

For any $k$, there exists a positive constant $b_{k}$ independent of $n$ so that

[TABLE] 2. (2).

For any $n$ and $k$, $u_{n}$ is continuous on $S_{\sigma_{k}}^{+}$. For any $k$, $u_{n}$ converges to $u_{\infty}$ uniformly on $S_{\sigma_{k}}^{+}$ as $n\rightarrow\infty$; 3. (3).

For any $n$ and $k$, $y_{n}+\epsilon_{n}\hskip 0.5ptu_{n}$ takes value in $\Gamma$ when it is evaluated at $\left(\hskip 1.0pt\pm\hskip 1.5pt\sigma_{k},0\hskip 1.0pt\right)$; 4. (4).

If I or II in (2.28) holds, then for any $k$, $u_{\infty}\left(\hskip 1.0pt\pm\hskip 1.5pt\sigma_{k},0\hskip 1.0pt\right)\in Y_{*}+\mathrm{Tan}_{y_{*}}\hskip 1.0pt\Gamma$; 5. (5).

If III in (2.28) holds, then for any $k$, $u_{\infty}\left(\hskip 1.0pt\pm\hskip 1.5pt\sigma_{k},0\hskip 1.0pt\right)\in Y_{*}+v_{-}+\mathrm{Tan}_{y_{*}}^{+}\hskip 1.0pt\Gamma$.

[TABLE]

Therefore it turns out \displaystyle\liminf_{n\rightarrow\infty}\int_{S_{\sigma}^{+}}\big{|}\hskip 1.0ptD_{\xi}u_{n}\hskip 1.0pt\big{|}^{2}<\infty, for $\mathscr{H}^{1}$-almost all $\sigma\in(0,1)$. In light that $u_{\infty}\in H^{1}(D^{+}_{1})$, we also have \displaystyle\int_{S_{\sigma}^{+}}\big{|}\hskip 1.0ptD_{\xi}u_{\infty}\hskip 1.0pt\big{|}^{2}<\infty, for $\mathscr{H}^{1}$-almost all $\sigma\in(0,1)$. By trace theorem, $y_{n}+\epsilon_{n}\hskip 0.5ptu_{n}\in\Gamma$ for $\mathscr{H}^{1}$-almost all $\xi\in T_{1}$. In light of all the arguments above and Lemmas 2.7-2.8, we can find an increasing positive sequence, denoted by \big{\{}\hskip 0.5pt\sigma_{k}\hskip 0.5pt\big{\}}, which takes limit $1$ as $k\rightarrow\infty$ and a subsequence of \big{\{}\hskip 0.5ptu_{n}\hskip 0.5pt\big{\}}, still denoted by \big{\{}u_{n}\big{\}}, so that (1) and (3)-(5) in Lemma 2.9 hold. Moreover it satisfies

[TABLE]

In the remaining we consider (2) in Lemma 2.9. Fix $n$ and $k$. By (1) in Lemma 2.9, $u_{n}$ is absolute continuous on $S_{\sigma_{k}}^{+}$. Meanwhile there holds, for all \alpha_{1},\alpha_{2}\in\big{[}\hskip 0.5pt0,\pi\hskip 0.5pt\big{]}, that

[TABLE]

This infers the equicontinuity of the family \left\{u_{n}\hskip 0.5pt\big{|}\hskip 0.5pt_{S_{\sigma_{k}}^{+}}:n\in\mathbb{N}\hskip 1.0pt\right\}. Using (2.35), we can find a \alpha_{0}\in\big{(}0,\pi\big{)} so that $u_{n}$ converges to $u_{\infty}$ at \big{(}\sigma_{k}\cos\alpha_{0},\hskip 1.0pt\sigma_{k}\sin\alpha_{0}\big{)}. Then by the last estimate, the $L^{\infty}$-norm of $u_{n}$ on $S_{\sigma_{k}}^{+}$ is uniformly bounded with respect to $n$. In light of Arzelà-Ascoli theorem, there exists a subsequence of \big{\{}u_{n}\big{\}} so that the subsequence uniformly converges to $u_{\infty}$ on $S_{\sigma_{k}}^{+}$. Applying inductive arguments and diagonal process, we can extract a subsequence from \big{\{}u_{n}\big{\}} so that (2) in Lemma 2.9 holds. The proof finishes. ∎

To proceed we need to introduce the concept of energy minimizing map in $\mathscr{F}_{A,\hskip 0.5pt1}$ and $\mathscr{F}_{B,\hskip 0.5pt1}$.

[TABLE]

Now we characterize the limiting function $u_{\infty}$ as follows:

Step 1. Comparison map. Let \big{\{}\hskip 1.0pt\sigma_{k}\hskip 1.0pt\big{\}} and \big{\{}\hskip 1.0ptu_{n}\hskip 1.0pt\big{\}} be the sequences in Lemma 2.9. Fixing a $k$, we pick up an arbitrary v\hskip 1.0pt\in\hskip 1.0ptH^{1}\big{(}\hskip 0.5ptD_{\sigma_{k}}^{+};Y_{*}+\mathrm{Tan}_{y_{*}}\hskip 1.0pt\mathbb{S}^{2}\hskip 1.0pt\big{)} with $v=u_{\infty}$ on $S_{\sigma_{k}}^{+}$. Moreover, in the sense of trace we let

[TABLE]

Here $v_{-}$ is given in Lemma 2.8. We cannot compare directly the energies between $v$ and $u_{n}$ since they have different ranges and boundary values.

Fixing an arbitrary $R>0$, we define for any $n$ the map

[TABLE]

In (2.37), a\vee b:=\mathrm{max}\big{\{}a,b\big{\}} for two real numbers $a$, $b$. The vector $v_{*}$ is [math] if I or II in (2.28) holds. It equals to $v_{-}$ if III in (2.28) holds. For large $n$, the length of $\mathscr{V}_{n}$ is close to $1$. Then we define

[TABLE]

Recall that the sequence \Big{\{}\hskip 1.0pt\Gamma_{n}\hskip 1.0pt\Big{\}}\subset\Gamma and converges to $y_{*}$, as $n\rightarrow\infty$. By taking $n$ sufficiently large and assuming I in (2.28), it turns out $v_{n}\hskip 2.0pt\in\hskip 2.0pt\Gamma$, for all points on $T_{\sigma_{k}}$ at which $v\in Y_{*}+\mathrm{Tan}_{y_{*}}\Gamma$. Similarly if $n$ is large and III in (2.28) holds, we also have $v_{n}\hskip 2.0pt\in\hskip 2.0pt\Gamma$, for all points on $T_{\sigma_{k}}$ at which $v\in Y_{*}+v_{-}+\mathrm{Tan}_{y_{*}}^{+}\Gamma$. As for II in (2.28), we decompose $\mathscr{V}_{n}$ into

[TABLE]

Denoting by $v^{+}$ the unit vector in $\mathrm{Tan}_{y_{*}}^{+}\Gamma$, we rewrite the above equality by

[TABLE]

When $n$ is large enough, $\Pi_{\Gamma}\left(y_{n}\right)$ and $\gamma_{n}$ are close to $y_{*}$. Hence it holds 2\hskip 1.0pt\big{|}\hskip 1.0pt\Pi_{\Gamma}\left(y_{n}\right)-\gamma_{n}\hskip 1.0pt\big{|}>\big{|}\hskip 1.0pt\Pi_{\Gamma}(y_{n})-y_{*}\hskip 1.0pt\big{|}, for large $n$. By this lower bound and the assumption in II of (2.28), we get

[TABLE]

In light that $v^{+}\in\mathrm{Tan}_{y_{*}}^{+}\Gamma$, then for all points on $T_{\sigma_{k}}$ with $v\in Y_{*}+\mathrm{Tan}_{y_{*}}\Gamma$, it turns out

[TABLE]

Utilizing (2.39)-(2.40) yields $v_{n}\in\Gamma$, for large $n$ and all points on $T_{\sigma_{k}}$ at where $v\in Y_{*}+\mathrm{Tan}_{y_{*}}\Gamma$. Here we also have used the limit \big{<}\hskip 1.0pt\Pi_{\Gamma}\left(y_{n}\right),\hskip 0.5pty_{*}\hskip 1.0pt\big{>}\longrightarrow 1, as $n\rightarrow\infty$. Since (2.36) holds in the sense of trace for all three cases in (2.28), the above arguments induce

[TABLE]

Similarly we define

[TABLE]

In light of (4)-(5) in Lemma 2.9, same arguments for the $v_{n}$ above infer

[TABLE]

Utilizing $v_{n}$ and $v_{n}^{\infty}$, for arbitrary $s\in\left(0,1\right)$, we define

[TABLE]

In light of (2) in Lemma 2.9, (2.25), (1) in (2.19), and the limit $\Pi_{\Gamma}\left(y_{n}\right)\rightarrow y_{*}$, the two sequences \Big{\{}\hskip 1.0ptu\left(x_{n}+r_{n}\hskip 1.0pt\cdot\right)\Big{\}} and \Big{\{}\hskip 1.0ptv_{n}^{\infty}\hskip 1.0pt\Big{\}} uniformly converge to $y_{*}$ on $S_{\sigma_{k}}^{+}$. Hence we can define

[TABLE]

For large $n$, it holds $u_{n}^{*}\in H^{1}\left(D_{\sigma_{k}}^{+};\hskip 1.0pt\mathbb{S}^{2}\right)$ with $u_{n}^{*}=u\left(x_{n}+r_{n}\hskip 1.0pt\cdot\right)$ on $S_{\sigma_{k}}^{+}$. By (2.43) and (3) in Lemma 2.9, we can take $n$ large enough and get

[TABLE]

Here we also have used the uniform convergence of \Big{\{}\hskip 1.0ptu\left(x_{n}+r_{n}\hskip 1.0pt\cdot\right)\Big{\}} and \Big{\{}\hskip 1.0ptv_{n}^{\infty}\hskip 1.0pt\Big{\}} to $y_{*}$ on $S_{\sigma_{k}}^{+}$. With (2.41), (2.44)-(2.46), it follows $u_{n}^{*}\in\Gamma$, for $\mathscr{H}^{1}$-almost all points on $T_{\sigma_{k}}$.

Step 2. Lower bound. Note that $u$ minimizes the $E_{\mathbb{D}}$-energy within the configuration space:

[TABLE]

By the minimality of $u$, it turns out

[TABLE]

Here $F_{2}=F_{2}(\rho,u):=\rho^{-2}\left(4u_{1}^{2}+u_{3}^{2}\right)+\sqrt{2}\mu\left(1-3P(u)\right)$. Up to a subsequence, as $n\rightarrow\infty$,

[TABLE]

By this limit, (2.17), (1) in (2.22) and lower-semi continuity, it follows

[TABLE]

In light of (2) in (2.19), it holds

[TABLE]

Applying this limit and (2.49) to the left-hand side of (2.47), we get

[TABLE]

Step 3. Upper bound. By the definition of $u_{n}^{*}$ in (2.45), it follows

[TABLE]

Firstly we estimate the right-hand side above. In light of (2.37), on $D_{\sigma_{k}}^{+}$, it satisfies

[TABLE]

With the definition of $v_{n}$ in (2.38) and the last convergence, for $j=1,2$, there holds

[TABLE]

Since $v-v_{*}-Y_{*}\in\mathrm{Tan}_{y_{*}}\mathbb{S}^{2}$ for almost all points in $D_{\sigma_{k}}^{+}$, the above limit infers

[TABLE]

By the polar coordinates $\left(\hskip 0.5pt\tau,\Theta\hskip 0.5pt\right)$ in the $\xi$-plane, the last integral in (2.52) can be represented as

[TABLE]

Using (2) in Lemma 2.9 and the definition (2.21), for any $\epsilon>0$, we can take $n$ large enough and get

[TABLE]

Still by (2) in Lemma 2.9, $u_{\infty}$ is continuous and uniformly bounded on $S_{\sigma_{k}}^{+}$. In light of $\mathscr{V}_{n}^{\infty}$ defined in (2.42), we can take $R$ large enough so that

[TABLE]

For I or II in (2.28), $\Gamma_{n}=\Pi_{\Gamma}\left(y_{n}\right)$ and $v_{*}=0$. Applying (2.25) then yields

[TABLE]

Here $o\left(\epsilon_{n}\right)$ denotes a quantity satisfying o\left(\epsilon_{n}\right)\big{/}\hskip 0.5pt\epsilon_{n}\rightarrow 0, as $n\rightarrow\infty$. For III in (2.28), $\Gamma_{n}=y_{*}$ and $v_{*}=v_{-}$. By the limits in (2.25) and (2.31), we also have

[TABLE]

Therefore in all cases, it holds

[TABLE]

Still by (2.56), we can calculate

[TABLE]

Since $u_{\infty}-v_{*}-Y_{*}\in\mathrm{Tan}_{y_{*}}\hskip 1.0pt\mathbb{S}^{2}$ on $S_{\sigma_{k}}^{+}$, then by $\Gamma_{n}\longrightarrow y_{*}$ as $n\rightarrow\infty$, the last equality can be reduced to

[TABLE]

which furthermore yields

[TABLE]

By (2.42) and (2.57)-(2.58), we get

[TABLE]

Hence for any $\epsilon>0$, the following estimate holds

[TABLE]

In light of this estimate and (2.55), by triangle inequality, we obtain

[TABLE]

Utilizing (2.44)-(2.45) and (2.61) above, for any $\epsilon>0$, we can take $n$ large enough and get

[TABLE]

Therefore it follows

[TABLE]

Using (2.44), (2.21), (2.42) and (2.56), on $D_{\sigma_{k}}^{+}\sim\hskip 1.0ptD_{\left(1-s\right)\sigma_{k}}^{+}$ we can rewrite $v_{n}^{*}$ as

[TABLE]

In light of (2.45) and the above representation of $v_{n}^{*}$, it then turns out

[TABLE]

Therefore by (1) in Lemma 2.9, we have

[TABLE]

Applying the above estimate and (2.62) to right-hand side of (2.54) yield

[TABLE]

Now we multiply $\epsilon_{n}^{-2}$ on both sides of (2.52) and take $n\rightarrow\infty$. By (2.53) and (2.63), it turns out

[TABLE]

Here $c>0$ is a constant. Moreover by taking $R\rightarrow\infty$, $\epsilon\rightarrow 0$ and $s\rightarrow 0$ successively, the above estimate induces

[TABLE]

Similarly as (2.50), it holds

[TABLE]

Applying this estimate and (2.64) to the right-hand side of (2.47) yield

[TABLE]

Step 4. Completion of proof. In light of (2.51) and (2.65), we can take $n\rightarrow\infty$ in (2.47) and obtain

[TABLE]

Since $\sigma_{k}\uparrow 1$ as $k\rightarrow\infty$, $u_{\infty}$ is an energy minimizing map in $\mathscr{F}_{A,\hskip 1.0pt1}$ if I or II in (2.28) holds. If III in (2.28) holds, then $u_{\infty}$ is an energy minimizing map in $\mathscr{F}_{B,\hskip 1.0pt1}$. Moreover if we take $v$ in (2.65) to be $u_{\infty}$, then by (2.47), (2.50)-(2.51) and (2.65), it holds $D_{\xi}\hskip 0.5ptu_{n}\rightarrow D_{\xi}\hskip 0.5ptu_{\infty}$ strongly in $L^{2}\left(D_{\sigma_{k}}^{+}\right)$. Still by the limit $\sigma_{k}\uparrow 1$ as $k\rightarrow\infty$, the strong $H^{1}\left(D_{r}^{+}\right)$-convergence of $u_{n}$ follows for any $r\in(0,1)$. ∎

With the above compactness result and characterization of $u_{\infty}$, we are ready to prove Lemma 2.2.

[TABLE]

Moreover utilizing (3) in (2.15), non-negativity of $F_{2}$ and change of variable, we have

[TABLE]

In light of the strong $H^{1}(D_{\theta_{1}}^{+})$-convergence of $u_{n}$ in Lemma 2.11, (2.50) and (2.66) above, we can take $n\rightarrow\infty$ in the last estimate and obtain

[TABLE]

For any $v\in H^{1}\left(D_{1}^{+};\hskip 1.0ptY_{*}+\mathrm{Tan}_{y_{*}}\hskip 1.0pt\mathbb{S}^{2}\right)$, there are two $H^{1}$ functions on $D_{1}^{+}$, denoted by $f^{v}_{1}$ and $f^{v}_{2}$, so that

[TABLE]

Here $v^{+}$ is the unit vector in $\mathrm{Tan}^{+}_{y_{*}}\hskip 1.0pt\Gamma$. Therefore by Lemma 2.11, if I or II in (2.28) holds, then it satisfies

[TABLE]

for any $f_{1}$, $f_{2}$ \in H^{1}\big{(}D_{1/2}^{+}\big{)} with $f_{1}=f_{1}^{u_{\infty}}$ on $S_{1/2}^{+}$ and $f_{2}=f_{2}^{u_{\infty}}$ on ${\partial}\hskip 0.5ptD_{1/2}^{+}$  . Notice that $f_{2}=0$ on $T_{1/2}$. Taking $f_{2}=f_{2}^{u_{\infty}}$ on the right-hand side of (2.68), we have

[TABLE]

Extending $f_{1}^{u_{\infty}}$ to the whole $D_{1/2}$ so that the extension, still denoted by $f_{1}^{u_{\infty}}$, is an even function with respect to the variable $\xi_{2}$, by the minimality of $f_{1}^{u_{\infty}}$ in (2.69), we can show that $f_{1}^{u_{\infty}}$ is a harmonic function on $D_{1/2}$. Applying Lemma 1.41 in [27] to $D_{\xi}\hskip 0.5ptf_{1}^{u_{\infty}}$ then yields

[TABLE]

Taking $f_{1}=f_{1}^{u_{\infty}}$ on the right-hand side of (2.68), we have

[TABLE]

Since $f_{2}^{u_{\infty}}=0$ on $T_{1/2}$, we can extend $f_{2}^{u_{\infty}}$ to the whole $D_{1/2}$ so that the extension, still denoted by $f_{2}^{u_{\infty}}$, is an odd function with respect to the variable $\xi_{2}$. By the minimality of $f_{2}^{u_{\infty}}$ given in (2.71), we can show that $f_{2}^{u_{\infty}}$ is a harmonic function on $D_{1/2}$. By Lemma 1.41 in [27], it turns out

[TABLE]

Utilizing (2.70), (2.72) and (2.66), we then obtain

[TABLE]

Therefore we can take $\theta_{1}$ suitably small so that (2.67) does not hold.

If III in (2.28) holds, then for any $v\in H^{1}\left(D_{1}^{+};\hskip 1.0ptY_{*}+\mathrm{Tan}_{y_{*}}\hskip 1.0pt\mathbb{S}^{2}\right)$, there are two $H^{1}$ functions on $D_{1}^{+}$, denoted by $f^{v}_{1}$ and $f^{v}_{2}$, so that

[TABLE]

Repeating the similar arguments for (2.69), we obtain

[TABLE]

Here we have used the boundary condition on $T_{3/4}$ for an admissible map in $\mathscr{F}_{B,\hskip 0.5pt3/4}$ (see (2.34)). Applying Lemma 9.1 in [53] to $f_{1}^{u_{\infty}}$, by Poincaré inequality, filling hole argument and standard iteration argument, we can find an universal constant \alpha\in\big{(}0,1\big{)} so that

[TABLE]

As for $f_{2}^{u_{\infty}}$, we can also extend it to be an odd function with respect to the variable $\xi_{2}$. (2.72) still holds in this case. By (2.72) and the last estimate, it turns out

[TABLE]

for any $\theta_{1}<3/16$. Therefore we can take $\theta_{1}$ suitably small so that (2.67) still fails in this case. ∎

**III. REGULARITY AT THE ORIGIN

**Due to Proposition 2.1, L\hskip 1.0pt\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]} and L\hskip 1.0pt\big{[}\hskip 1.0ptu_{c}^{-}\hskip 1.0pt\big{]} solve the equation in (1.13) weakly in a neighborhood of [math]. Applying Rivi$\grave{\text{e}}$re-Struwe arguments in [54], we obtain the following $\epsilon$-regularity near the origin:

[TABLE]

As a map defined on $B_{1}$, $u_{\hskip 1.0ptb}^{+}$ is also regular at the origin in the sense of Definition 1.2, if the map $\mathcal{W}_{\hskip 1.0ptb}^{+}$ is smooth at the origin. The same result holds for $\mathcal{W}_{\hskip 1.0ptc}^{-}$ and $u_{\hskip 1.0ptc}^{-}$.

In this section we improve the result in Proposition 3.1 by showing

Throughout the section we use $\mathcal{W}$ to denote either $\mathcal{W}_{\hskip 1.0ptb}^{+}$ or $\mathcal{W}_{\hskip 1.0ptc}^{-}$. To prove Proposition 3.2, we need to show that all tangent maps of $\mathcal{W}$ at [math] are trivial. Since $u_{b}^{+}$ and $u_{c}^{-}$ are minimizers of the two minimization problems in (1.12), by applying arguments in [21], for almost all $R\in(0,1)$, it holds

[TABLE]

Here ${\partial}^{+}B_{R}$ denotes the spherical boundary of $B_{R}^{+}$. $\nu$ is the outer normal direction on ${\partial}^{+}B_{R}$. Utilizing the last energy equality then yields

[TABLE]

Supposing that \Big{\{}k_{j}\Big{\}} is a positive decreasing sequence which converges to [math] as $j\rightarrow\infty$, we rescale $\mathcal{W}$ by

[TABLE]

The map $\mathcal{W}^{(j)}$ is well-defined, provided $j$ is large. Moreover by (3.1), it turns out

[TABLE]

The limiting map $\mathcal{W}^{\infty}$ is a [math]-homogeneous harmonic map from $B_{1}$ to $\mathbb{S}^{4}$ with energy satisfying

[TABLE]

Moreover in the sense of trace, there holds

[TABLE]

Here $T$ is the flat boundary of $B^{+}_{1}$. $\mathcal{W}_{3}^{\infty}$ is the third component of $\mathcal{W}^{\infty}$.

[TABLE]

Utilizing this uniform energy bound then yields (3.3). Moreover by lower semi-continuity, we can take $j\rightarrow\infty$ in the last estimate and get

[TABLE]

Since $\mathcal{W}=L\left[\hskip 0.5ptu\hskip 0.5pt\right]$ with $u$ equaling to either $u_{\hskip 1.0ptb}^{+}$ or $u_{\hskip 1.0ptc}^{-}$, then by (1.12), the last estimate can be reduced to

[TABLE]

(3.4) holds by the last estimate. As for (3.5), we notice that the trace of $\mathcal{W}_{3}$ on $T$ equals to the trace of $u_{2}$ on $T$. It then turns out

[TABLE]

By Theorem 6.2 in [50], we can keep extracting a subsequence, still denoted by $\left\{\mathcal{W}^{(j)}\right\}$, so that

[TABLE]

Utilizing (3.7), we can take $j\rightarrow\infty$ in (3.6) and obtain (3.5). The fact that $\mathcal{W}^{\infty}$ is a [math]-homogeneous harmonic map from $B_{1}$ to $\mathbb{S}^{4}$ follows by a standard argument in [56]. ∎

Since $\mathcal{W}^{\infty}$ is a [math]-homogeneous harmonic map from $B_{1}$ to $\mathbb{S}^{4}$, $\mathcal{W}^{\infty}$ can be represented by $\mathcal{W}^{\infty}=L\left[\hskip 0.5ptv\hskip 0.5pt\right]$, where $v=(v_{1},v_{2},v_{3})$ depends only on the polar angle $\varphi$ and is a smooth $3$-vector field on \big{[}\hskip 0.5pt0,\pi\hskip 0.5pt\big{]} with unit length. Moreover the three components of $v$ satisfy the following ODE system:

[TABLE]

Here ′ denotes the derivative with respect to the variable $\varphi$. Utilizing the system (3.8), we have

[TABLE]

Since $v$ is smooth on \big{[}\hskip 0.5pt0,\pi\hskip 0.5pt\big{]} and $v_{1}=v_{3}=0$ at $\varphi=0$, $\mathrm{const}$ in the last equality equals to [math]. It turns out

[TABLE]

Note that $v_{1}\left(\varphi\right)=v_{1}\left(\pi-\varphi\right)$, $v_{2}\left(\varphi\right)=v_{2}\left(\pi-\varphi\right)$, $v_{3}\left(\varphi\right)=-v_{3}\left(\pi-\varphi\right)$, for any \varphi\in\big{[}\hskip 0.5pt0,\pi\hskip 0.5pt\big{]}. Then

[TABLE]

Evaluating (3.9) at $\varphi=\dfrac{\pi}{2}$ and using (3.10), we obtain

[TABLE]

Here we also used the boundary condition $v_{3}=0$ at $\varphi=\pi/2$. Since

[TABLE]

it holds $v_{3}^{\prime}\leq 0$ at $\pi/2$. The sign in (3.11) can be determined. That is

[TABLE]

Suppose that $v_{1}=0$ at some $\varphi_{0}\in(\hskip 0.5pt0,\pi/2\hskip 0.5pt)$. By (3.12), $\varphi_{0}$ is a minimum point of $v_{1}$ on $(\hskip 0.5pt0,\pi/2\hskip 0.5pt)$. It follows $v_{1}^{\prime}=0$ at $\varphi_{0}$. Applying uniqueness result for ordinary differential equation to the equation of $v_{1}$ in (3.8) then yields $v_{1}\equiv 0$ on $\left[\hskip 0.5pt0,\pi\hskip 0.5pt\right]$. In light of (3.13), $v_{3}^{\prime}=0$ at $\varphi=\pi/2$. Since we have $v_{3}=0$ at $\varphi=\pi/2$, we can also apply uniqueness result for ordinary differential equation to the equation of $v_{3}$ in (3.8) and obtain $v_{3}\equiv 0$ on $\left[\hskip 0.5pt0,\pi\hskip 0.5pt\right]$. In this case we have $\mathcal{W}^{\infty}$ equals to either $N^{*}$ or $-N^{*}$ on $B_{1}$.

Similarly if it holds $v_{3}=0$ at some $\varphi_{0}\in(\hskip 0.5pt0,\pi/2\hskip 0.5pt)$, then we have $v_{3}\equiv 0$ on $\left[\hskip 0.5pt0,\pi\hskip 0.5pt\right]$. Still by (3.13), it follows $v_{1}=0$ at $\varphi=\pi/2$. From (3.10), it satisfies $v_{1}^{\prime}=0$ at $\varphi=\pi/2$. We then can still infer $v_{1}\equiv 0$ on $\left[\hskip 0.5pt0,\pi\hskip 0.5pt\right]$. In this case $\mathcal{W}^{\infty}$ also equals to either $N^{*}$ or $-N^{*}$ on $B_{1}$.

Now we assume $v_{1}>0$ and $v_{3}>0$ on $(\hskip 0.5pt0,\pi/2\hskip 0.5pt)$. Since $v_{2}=1$ or $-1$ at $\varphi=0$, by the regularity of $v_{2}$ and the positivity of $v_{1}$ and $v_{3}$ on $(\hskip 0.5pt0,\pi/2\hskip 0.5pt)$, we can find a $\varphi_{1}\in(\hskip 0.5pt0,\pi/2\hskip 0.5pt)$ so that $v_{1}$, $v_{2}$ and $v_{3}$ are all non-zero at $\varphi_{1}$. Therefore the image of $\mathcal{W}^{\infty}$ restricted on \big{(}\varphi_{1},\theta\hskip 0.5pt\big{)}, where $\theta\in\left[\hskip 0.5pt0,2\pi\hskip 0.5pt\right]$, can not be contained in a totally geodesic $\mathbb{S}^{3}\subset\mathbb{S}^{4}$. Here we also used $\mathcal{W}^{\infty}=L\hskip 1.0pt\left[\hskip 1.0ptv\hskip 1.0pt\right]$ and the definition of the $L$-operator. This infers that $\mathcal{W}^{\infty}$, when restricted on $\mathbb{S}^{2}$, is a linearly full harmonic map from $\mathbb{S}^{2}$ to $\mathbb{S}^{4}$. By Corollary 1.20 in [39], the twistor degree $d$ of $\mathcal{W}^{\infty}$ must satisfy $d\geq 3$. It follows $\displaystyle\int_{B_{1}}\left|\hskip 1.0pt\nabla\mathcal{W}^{\infty}\hskip 1.0pt\right|^{2}\hskip 2.0pt=\hskip 2.0pt8\hskip 0.5pt\pi\hskip 0.5ptd\hskip 2.0pt\geq\hskip 2.0pt24\hskip 1.0pt\pi$. This estimate violates (3.4).

Hence $\mathcal{W}^{\infty}=N^{*}$ or $-N^{*}$. The lemma then follows by (3.5). ∎

**IV. SINGULARITY: ITS TANGENT MAP AND STRUCTURE

**Denote by $x_{0}$ the point \big{(}\hskip 0.5pt0,0,z_{0}\hskip 0.5pt\big{)}, where $z_{0}\in(0,1)$. Let \big{\{}k_{j}\big{\}} be a positive decreasing sequence which converges to [math] as $j\rightarrow\infty$. If $j$ is sufficiently large, then the map

[TABLE]

is well-defined. Here we still use $\mathcal{W}$ to denote either L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptb}^{+}\hskip 0.5pt\big{]} or L\hskip 1.0pt\big{[}\hskip 0.5ptu_{\hskip 1.0ptc}^{-}\hskip 0.5pt\big{]}. Similar arguments as the proof for (3.3) implies that there exists a $w^{\infty}\in H^{1}\left(B_{1};\mathbb{S}^{4}\right)$ so that up to a subsequence

[TABLE]

Moreover $w^{\infty}$ is a [math]-homogeneous harmonic map from $B_{1}$ to $\mathbb{S}^{4}$. By a similar construction as in the proof of convergence theorem 5.5 in [31], it holds, for all $R\in(0,1)$, that

[TABLE]

On the other hand lower semi-continuity and (1) in (4.2) infers

[TABLE]

Combining the last lower bound and (4.3), we get $w^{(j)}\longrightarrow w^{\infty}$, strongly in $H^{1}\left(B_{R};\mathbb{S}^{4}\right)$, as $j\rightarrow\infty$. Since $R$ is an arbitrary number in $(0,1)$, it then holds

[TABLE]

**IV.1. MINIMALITY OF THE LIMITING MAP

**In this section we study the minimality of $w^{\infty}$ with respect to the Dirichlet energy. The main result is

[TABLE]

Before proving Proposition 4.1, we introduce some notations. Given a $3$-vector field u=\big{(}u_{1},u_{2},u_{3}\big{)} on \big{[}\hskip 0.5pt0,\pi\hskip 0.5pt\big{]}, we use $e_{2}\left[\hskip 0.5ptu\hskip 0.5pt\right]$ to denote the energy density $\left|{\partial}_{\varphi}u\hskip 1.0pt\right|^{2}\sin\varphi+\dfrac{1}{\sin\varphi}\left(4u_{1}^{2}+u_{3}^{2}\right)$. The $\mathrm{E}_{2}$-energy of $u$ is the integration of $e_{2}\left[\hskip 0.5ptu\hskip 0.5pt\right]$ on \big{[}\hskip 0.5pt0,\pi\hskip 0.5pt\big{]}. Firstly we control the $L^{\infty}$-norm of $u$ in terms of its $\mathrm{E}_{2}$-energy. That is

[TABLE]

If in addition $|u\hskip 0.5pt|=1$ on \big{[}\hskip 0.5pt0,\pi\hskip 0.5pt\big{]}, then $u_{2}$ satisfies

[TABLE]

The proof is standard, we omit it here. Now we prove Proposition 4.1.

[TABLE]

Then up to a subsequence,

[TABLE]

Utilizing (4.6) and the limit (4.8), we obtain

[TABLE]

Let $\varphi_{*}$ be an arbitrary number in $(\hskip 0.5pt0,\pi/2)$. By (4.7), it follows

[TABLE]

In light of (4.8)-(4.9) and taking $j\rightarrow\infty$ in the last estimate, we then obtain

[TABLE]

The last two estimates then yield

[TABLE]

Here and in what follows, for a function defined on an interval $I$, we use $\|\cdot\|_{\infty;\hskip 1.0ptI}$ to denote its $L^{\infty}$-norm on the interval $I$. Similarly it also holds

[TABLE]

As for \varphi_{1}\in\big{[}\hskip 0.5pt\varphi_{*},\hskip 1.0pt\pi-\varphi_{*}\hskip 0.5pt\big{]}, we have the lower bound $\sin\varphi_{1}\geq\sin\varphi_{*}$. It induces

[TABLE]

Therefore it holds

[TABLE]

Using the last estimate, (4.11)-(4.12) and the limits in (4.8)-(4.9), we get

[TABLE]

By (4.10) and the above limit, it turns out

[TABLE]

Suppose that $w^{*}=L\left[\hskip 0.5ptu\hskip 0.5pt\right]$ for some $u=u\left(r,\varphi\right)$ on $B_{R}$. Moreover we assume $w^{*}\in H^{1}\left(B_{R};\hskip 1.0pt\mathbb{S}^{4}\right)$ and satisfies, in the sense of trace, $w^{*}=w^{\infty}$ on ${\partial}B_{R}$. We need to show

[TABLE]

Firstly we extend $w^{*}$ from $B_{R}$ to $B_{\sigma_{2}}$ by setting

[TABLE]

Fixing a \sigma_{1}\in\big{(}R,\sigma_{2}\big{)}, we define

[TABLE]

In light of (4.13), the distance between $\widetilde{w}^{(j)}$ and $\mathbb{S}^{4}$ is uniformly small on $B_{\sigma_{2}}$, provided $j$ is sufficiently large. Therefore we can define our comparison map

[TABLE]

Moreover it holds

[TABLE]

By (4.16), the energy of $\widetilde{w}^{(j)}$ on $B_{\sigma_{2}}\sim B_{\sigma_{1}}$ can be estimated by

[TABLE]

Applying the last estimate to (4.18) and taking $j$ large, we obtain

[TABLE]

Since $u_{b}^{+}$ and $u_{c}^{-}$ are minimizers of the two minimization problems in (1.12), respectively,

[TABLE]

By the boundedness of potential term and (4.15)-(4.17), the last estimate infers

[TABLE]

Here $c_{\mu}$ is a positive constant depending only on $\mu$. Utilizing (4.19) and convergence in (4.4), (4.8) and (4.13), we can take $j\rightarrow\infty$ in the last estimate and get

[TABLE]

Here $c>0$ is a universal constant. (4.14) then follows by taking $\sigma_{1}\rightarrow\sigma_{2}^{-}$ in the above estimate. ∎

**IV.2. GENERIC SINGULARITIES

**Since w^{\infty}=L\hskip 0.5pt\big{[}\hskip 0.5ptu^{\infty}\hskip 0.5pt\big{]} is a [math]-homogeneous harmonic map from $B_{1}$ to $\mathbb{S}^{4}$, the three components of $u^{\infty}$ satisfy the ODE system (3.8). By this ODE system, there holds

$\mathbf{Class\hskip 4.0ptI:}$  Constant mapping $N^{*}$ or $-N^{*}$, where $N^{*}=(0,0,1,0,0)$;

$\mathbf{Class\hskip 4.0ptII:}$  For some \beta_{1}\in\big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}, w^{\infty}=L\hskip 0.5pt\big{[}\hskip 0.5ptu^{\infty}\hskip 0.5pt\big{]} with $u^{\infty}$ equaling to*

[TABLE]

$\mathbf{Class\hskip 4.0ptIII:}$*  For some \beta_{2}\in\big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}, w^{\infty}=L\hskip 0.5pt\big{[}\hskip 0.5ptu^{\infty}\hskip 0.5pt\big{]} with $u^{\infty}$ equaling to*

[TABLE]

Here $J\left(\varphi\right):=\arccos\left(\dfrac{2\cos\varphi}{1+\cos^{2}\varphi}\right)$.

[TABLE]

If $v_{1}=0$ at some \varphi_{0}\in\big{(}0,\pi\big{)}, then by (4.20), $\varphi_{0}$ is a minimum point of $v_{1}$. It follows $v_{1}^{\prime}\left(\varphi_{0}\right)=0$. Here and in the following ′ denotes the derivative with respect to the variable $\varphi$. By applying uniqueness result for ordinary differential equation to the equation of $v_{1}$ in (3.8), it turns out $v_{1}\equiv 0$ on \big{[}\hskip 0.5pt0,\pi\hskip 0.5pt\big{]}. If in addition $v_{3}=0$ at some point in \big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}, then similarly we have $v_{3}\equiv 0$ on \big{[}\hskip 0.5pt0,\pi\hskip 0.5pt\big{]}. Here we just need to use the equation of $v_{3}$ in (3.8). In this case, $w^{\infty}$ is the constant mapping $N^{*}$ or $-N^{*}$. If $v_{1}\equiv 0$ on \big{[}\hskip 0.5pt0,\pi\hskip 0.5pt\big{]} and $v_{3}>0$ on \big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}, then we can represent $v$ by

[TABLE]

Plugging the representation (4.21) into (3.9), we get

[TABLE]

The sign of $\alpha^{\prime}$ does not change on \big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}. Otherwise there will be a \varphi_{1}\in\big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)} so that $\alpha^{\prime}(\varphi_{1})=0$. Using the last equality yields $\sin\alpha(\varphi_{1})=0$. It violates the assumption that $v_{3}\left(\varphi_{1}\right)=\sin\alpha(\varphi_{1})>0$. Therefore we have from the last equality that

[TABLE]

Solving the two ODEs in (4.22), we obtain the harmonic maps given in Class II of the lemma.

In the remaining we suppose $v_{1}>0$ on \big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}. In this case we multiply $v_{3}$ and $v_{1}$ on both sides of the equations of $v_{1}$ and $v_{3}$ in (3.8), respectively. Then we subtract one of the two resulting equations from another one. By this way, it turns out

[TABLE]

Here we also have used (4.20) in the last inequality above. (4.23) infers that \big{(}v_{1}v_{3}^{\prime}-v_{3}v_{1}^{\prime}\big{)}\hskip 1.0pt\sin\varphi is non-increasing on \big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}. The quantity \big{(}v_{1}v_{3}^{\prime}-v_{3}v_{1}^{\prime}\big{)}\sin\varphi equals to [math] when $\varphi=0$ and $\pi$. Therefore we have $\left(v_{1}v_{3}^{\prime}-v_{3}v_{1}^{\prime}\right)\sin\varphi\equiv 0$, for all \varphi\in\big{(}0,\pi\big{)}. Since $v_{1}>0$ on \big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}, it follows

[TABLE]

Therefore $v_{3}=cv_{1}$ on \big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}, where $c$ is a constant. If $c\neq 0$, then by the equations of $v_{1}$ and $v_{3}$ in (3.8), it holds $v_{1}\equiv 0$ on \big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}. It violates our assumption that $v_{1}>0$ on \big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}. Therefore $c=0$. Equivalently $v_{3}\equiv 0$ on \big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}. Now we can represent $v$ by v=\big{(}v_{1},v_{2},0\big{)}=\big{(}\sin\alpha,\hskip 1.0pt\cos\alpha,0\big{)}, where $\alpha=\arccos v_{2}$. Plugging this representation of $v$ into (3.9) then yields

[TABLE]

The sign of $\alpha^{\prime}$ does not change on \big{(}\hskip 0.5pt0,\pi\hskip 0.5pt\big{)}. Therefore we have from the last equality that

[TABLE]

Solving the two ODEs in (4.24), we obtain the harmonic maps given in Class III of the lemma. ∎

With Proposition 4.1 and Lemma 4.3, we can further determine $w^{\infty}$ at possible singularity $x_{0}$.

[TABLE]

If we denote by \mathrm{Hess}^{\hskip 1.0ptD}_{\hskip 1.0ptw^{\infty}}\big{[}\hskip 0.5pt\cdot,\cdot\hskip 0.5pt\big{]} the Hessian of Dirichlet energy at $w^{\infty}$, it then turns out

[TABLE]

By (4.25) and in light that $\alpha_{\star}$ satisfies either the first or the second equation in (4.24), it follows

[TABLE]

Using Lemma 1.3 in [58], we can find a $f$ compactly supported in the interval \big{(}0,1\big{)} so that the most-right-hand side above is strictly negative. The proof then completes. ∎

**IV.3. STRUCTURE OF SINGULARITY

**By the same arguments as in the proof of Regularity theorem 6.4 in [31], it holds

In this section we consider the asymptotic behaviour of $\mathcal{W}$ near each singularity. Firstly we show the uniqueness of tangent map of $\mathcal{W}$ at a possible singularity. That is

[TABLE]

where $\Lambda=\Lambda^{+}$ or $\Lambda^{-}$ in (1.14). Moreover $\Lambda$ is uniquely determined by $x_{0}$, the location of singularity.

[TABLE]

Here $N^{*}=(0,0,1,0,0)$. In light of Lemma 4.5, when $j$ is large enough, $x_{0}$ is the only singularity of $\mathcal{W}$ on the segment connecting $x_{j}^{+}$ and $x_{0}$. Therefore $\mathcal{W}\equiv N^{*}$ on the open segment between $x_{j}^{+}$ and $x_{0}$. Similarly for $j$ large, $\mathcal{W}\equiv-N^{*}$ on the open segment between $x_{j}^{-}$ and $x_{0}$. Fix a $j$ sufficiently large. If we have another positive decreasing sequence \big{\{}\rho_{k}\big{\}} so that up to a subsequence $\mathcal{W}\left(x_{0}+\rho_{k}\hskip 0.5pt\cdot\hskip 1.0pt\right)$ converges to $\Lambda^{-}$ strongly in H_{\mathrm{loc}}^{1}\big{(}B_{1};\hskip 0.5pt\mathbb{S}^{4}\big{)} as $k\rightarrow\infty$ (here $\rho_{k}\rightarrow 0$ as $k\rightarrow\infty$), then the same derivation of (4.26) can be applied to find a point on the open segment between $x_{j}^{+}$ and $x_{0}$ at which $\mathcal{W}$ equals to $-N^{*}$. But this is impossible. Therefore for any decreasing sequence \big{\{}\rho_{k}\big{\}}, there is a subsequence, still denoted by \big{\{}\hskip 0.5pt\rho_{k}\hskip 0.5pt\big{\}}, so that $\mathcal{W}\left(x_{0}+\rho_{k}\hskip 0.5pt\cdot\hskip 1.0pt\right)$ converges $\Lambda^{+}$ strongly in H_{\mathrm{loc}}^{1}\big{(}B_{1};\hskip 0.5pt\mathbb{S}^{4}\big{)} as $k\rightarrow\infty$. The proof then follows. ∎

Moreover the following proposition holds by Lemma 4.6, $W^{2,p}$-estimate for elliptic equations and similar arguments as in Sect.3.15 of [60].

[TABLE]

Here $\|\cdot\|_{\infty;\hskip 1.0pt{\partial}B_{\sigma}\left(x_{0}\right)}$ is the $L^{\infty}$-norm on ${\partial}B_{\sigma}\left(x_{0}\right)$.

**V. BIAXIAL TORUS AND SPLIT-CORE SEGMENT STRUCTURES

**In this section we prove Theorems 1.6-1.7. Recall that \mathcal{W}_{\hskip 1.0ptb}^{+}=L\hskip 1.0pt\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]} and \mathcal{W}_{\hskip 1.0ptc}^{-}=L\hskip 1.0pt\big{[}\hskip 1.0ptu_{c}^{-}\hskip 1.0pt\big{]}.

**V.1. BIAXAIL TORUS STRUCTURE IN $\mathcal{W}_{b}^{+}$

**To simplify the notation, we use $u^{+}$ to denote $u_{\hskip 1.0ptb}^{+}$. In this section $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ are three eigenvalues in Definition 1.3 computed in terms of $u^{+}$. Note that $u_{3}^{+}=0$ on $T$. By Definition 1.3, it follows $\lambda_{1}=-\dfrac{1}{2}\left(u_{1}^{+}+\dfrac{1}{\sqrt{3}}\hskip 1.0ptu_{2}^{+}\right)$, \lambda_{2}=\dfrac{1}{4}\bigg{\{}u_{1}^{+}+\dfrac{1}{\sqrt{3}}\hskip 1.0ptu_{2}^{+}-\left|\hskip 1.0ptu_{1}^{+}-\sqrt{3}\hskip 0.5ptu_{2}^{+}\hskip 1.0pt\right|\hskip 1.0pt\bigg{\}}, \lambda_{3}=\dfrac{1}{4}\bigg{\{}u_{1}^{+}+\dfrac{1}{\sqrt{3}}\hskip 1.0ptu_{2}^{+}+\left|\hskip 1.0ptu_{1}^{+}-\sqrt{3}\hskip 0.5ptu_{2}^{+}\hskip 1.0pt\right|\hskip 1.0pt\bigg{\}} on $T$. The three eigenvalues cannot be all [math] at some point on $T$, since otherwise $u^{+}=0$ at this point. If $\lambda_{1}=\lambda_{2}$ on $T$, then $3\left(u_{1}^{+}+\dfrac{1}{\sqrt{3}}\hskip 1.0ptu_{2}^{+}\right)=\left|\hskip 1.0ptu_{1}^{+}-\sqrt{3}\hskip 0.5ptu_{2}^{+}\hskip 1.0pt\right|$. If $\lambda_{2}=\lambda_{3}$ on $T$, then $u_{1}^{+}=\sqrt{3}\hskip 1.0ptu_{2}^{+}$. If $\lambda_{1}=\lambda_{3}$ on $T$, then $3\left(u_{1}^{+}+\dfrac{1}{\sqrt{3}}\hskip 1.0ptu_{2}^{+}\right)=-\left|\hskip 1.0ptu_{1}^{+}-\sqrt{3}\hskip 0.5ptu_{2}^{+}\hskip 1.0pt\right|$. Therefore by the unit length condition for $u^{+}$ and the strict positivity of $u_{1}^{+}$ (this is true by strong maximum principle), the map $u^{+}$ is uniaxial on T\sim\big{\{}\hskip 0.5pt0\hskip 0.5pt\big{\}} if and only if u^{+}=\big{(}\sqrt{3}\big{/}2,\hskip 1.0pt1\big{/}2,\hskip 1.0pt0\big{)} or u^{+}=\big{(}\sqrt{3}\big{/}2,\hskip 1.0pt-1\big{/}2,\hskip 1.0pt0\big{)}. In light that $u_{2}^{+}\geq b$ on $T$ and $u^{+}$ is regular at the origin (see Proposition 3.2), it follows $u_{2}^{+}=1$ at the origin. Since $u_{2}^{+}=-1/2$ when $\rho=1$ and $z=0$, using regularity of $u^{+}$ on $T$, we can find a point on $T$ at where $u_{2}^{+}=1/2$. Equivalently the vector \big{(}\sqrt{3}\big{/}2,\hskip 1.0pt1\big{/}2,\hskip 1.0pt0\big{)} can be taken by $u^{+}$ at some point on $T$. $u^{+}$ is a classical solution to (2.1) on a small neighborhood of ${\kern 0.0ptT}^{\mathrm{o}}$ in $\mathbb{D}$. Moreover the system (2.1) is an analytic elliptic system. Therefore due to [48], $u^{+}$ is real analytic on ${\kern 0.0ptT}^{\mathrm{o}}$, which infers real analyticity of $u_{2}^{+}$ on ${\kern 0.0ptT}^{\mathrm{o}}$. It then turns out that $1/2$ can only be taken finitely many times by $u_{2}^{+}$ on ${\kern 0.0ptT}^{\mathrm{o}}$. Otherwise $u_{2}^{+}\equiv 1/2$ on $T$. This is impossible. Since $u_{2}^{+}(0)=1$ and $u_{2}^{+}\left(1,0\right)=-1/2$, within all points on ${\kern 0.0ptT}^{\mathrm{o}}$ at where $u_{2}^{+}=1/2$, there is one, denoted by $x_{0}=\left(\rho_{0},0\right)$, so that for a sufficiently small $\epsilon>0$, it holds

[TABLE]

For points in $D_{\epsilon}\left(x_{0}\right)\sim T$, strong maximum principle induces that $u_{3}^{+}>0$ on $D_{\epsilon}\left(x_{0}\right)\cap\mathbb{D}^{+}$ and $u_{3}^{+}<0$ on $D_{\epsilon}\left(x_{0}\right)\cap\mathbb{D}^{-}$. Here $\mathbb{D}^{-}$ is the lower-half part of $\mathbb{D}$. By Definition 1.3, we then have

[TABLE]

It can also be computed that \lambda_{2}-\lambda_{1}\hskip 2.0pt=\hskip 2.0pt\dfrac{1}{4}\bigg{\{}3\left(u_{1}^{+}+\dfrac{1}{\sqrt{3}}\hskip 1.0ptu_{2}^{+}\right)-\sqrt{\left(u_{1}^{+}-\sqrt{3}\hskip 1.0ptu_{2}^{+}\right)^{2}+4\hskip 1.0pt\left(u_{3}^{+}\right)^{2}}\hskip 2.0pt\bigg{\}}. By the regularity of $u^{+}$ on $D_{\epsilon}\left(x_{0}\right)$ and the fact that u^{+}\left(x_{0}\right)=\left(\sqrt{3}\big{/}2,\hskip 1.0pt1\big{/}2,\hskip 1.0pt0\right), it then turns out

[TABLE]

This inequality and (5.1) infer

[TABLE]

On \Big{\{}\left(\rho,0\right):\rho\in\left(\hskip 0.5pt\rho_{0},\hskip 1.0pt\rho_{0}+\epsilon\right)\Big{\}}, it holds $u_{2}^{+}\in\left(\hskip 0.5pt0,1/2\right)$, provided $\epsilon$ is small. Therefore we have u_{1}^{+}>\sqrt{3}\big{/}2 on \Big{\{}\left(\rho,0\right):\rho\in\left(\hskip 0.5pt\rho_{0},\hskip 1.0pt\rho_{0}+\epsilon\right)\Big{\}}, which infers

[TABLE]

It then follows by Definition 1.3 that $\lambda_{3}\hskip 2.0pt>\hskip 2.0pt\lambda_{2}$ on \Big{\{}\left(\rho,0\right):\rho\in\left(\hskip 0.5pt\rho_{0},\hskip 1.0pt\rho_{0}+\epsilon\right)\Big{\}}, provided $\epsilon$ is small. Utilizing (5.2), we then obtain

[TABLE]

Similarly we also have

[TABLE]

By combining the above arguments with (5.3), on D_{\epsilon}\left(x_{0}\right)\sim\big{\{}x_{0}\big{\}}, $u^{+}$ is biaxial with $\lambda_{3}$ being the largest eigenvalue.

By (5.3), $\kappa^{\star}$ in (5) of Theorem 1.6 is a director field determined by $u^{+}$ on $D_{\epsilon}\left(x_{0}\right)\sim T$. It is also well-defined and continuous on \Big{\{}\left(\rho,0\right):\rho\in\left(\hskip 0.5pt\rho_{0},\hskip 1.0pt\rho_{0}+\epsilon\right)\Big{\}}. In fact by (5.4) and $u_{3}^{+}=0$ on $T$, it holds

[TABLE]

However $\kappa^{\star}$ is not continuous on \Big{\{}\left(\hskip 0.5pt\rho,0\hskip 0.5pt\right):\rho\in\big{(}\hskip 0.5pt\rho_{0}-\epsilon,\hskip 1.0pt\rho_{0}\big{]}\hskip 1.0pt\Big{\}}. Fix an $\epsilon^{\prime}\in\left(\hskip 0.5pt0,\epsilon\right)$ and denote by $x^{\prime}$ the point $\left(\hskip 0.5pt\rho_{0}-\epsilon^{\prime},\hskip 1.0pt0\hskip 0.5pt\right)$. When we approach $x^{\prime}$ along the lower part of ${\partial}D_{\epsilon^{\prime}}\left(x_{0}\right)$, the component $u_{3}^{+}$ keeps negative. It then follows

[TABLE]

where the left-hand side above is evaluated on the lower part of ${\partial}D_{\epsilon^{\prime}}\left(x_{0}\right)$. Since we have u_{2}^{+}>1\big{/}2 at $x^{\prime}$, it follows u_{1}^{+}<\sqrt{3}\big{/}2 at $x^{\prime}$. Therefore we get $u_{1}^{+}-\sqrt{3}\hskip 1.0ptu_{2}^{+}<0$ at $x^{\prime}$. Utilizing this inequality and the last equality, we can approach $x^{\prime}$ along the lower part of ${\partial}D_{\epsilon^{\prime}}\left(x_{0}\right)$ and get

[TABLE]

Meanwhile it holds

[TABLE]

The last two limits infer that when we approach $x^{\prime}$ along the lower part of ${\partial}D_{\epsilon^{\prime}}\left(x_{0}\right)$, the director field $\kappa^{\star}$ approaches to $-e_{z}$. Similarly when we approach $x^{\prime}$ along the upper part of ${\partial}D_{\epsilon^{\prime}}\left(x_{0}\right)$, the director field $\kappa^{\star}$ approaches to $e_{z}$. Here we need the positivity of $u_{3}^{+}$ on the upper part of ${\partial}D_{\epsilon^{\prime}}\left(x_{0}\right)$. By (1.15), $\kappa^{\star}$ has positive coefficient in front of $e_{\rho}$ when x\in{\partial}D_{\epsilon^{\prime}}\left(x_{0}\right)\sim\big{\{}\hskip 0.5ptx^{\prime}\hskip 0.5pt\big{\}}. Therefore when we start from $x^{\prime}$ and rotate counter-clockwisely along ${\partial}D_{\epsilon^{\prime}}\left(x_{0}\right)$ back to $x^{\prime}$, the director field $\kappa^{\star}$ continuously varies from $-e_{z}$ to $e_{z}$. Meanwhile $\kappa^{\star}$ keeps on the right-half part of $\left(\rho,z\right)$ - plane. The angle of $\kappa^{\star}$ is totally changed by $\pi$. Part $(6)$ in Theorem 1.6 follows by (5.5) and L’H$\mathrm{\hat{o}}$pital’s rule.

**V.2. SPLIT-CORE LINE SEGMENT STRUCTURE

**In this section we prove Theorem 1.7 by considering the mapping $\mathcal{W}_{\hskip 1.0ptc}^{-}$, equivalently $u_{\hskip 1.0ptc}^{-}$. To simplify the notation, we use $u^{-}$ to denote $u_{\hskip 1.0ptc}^{-}$. Throughout this section $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ are three eigenvalues in Definition 1.3 computed in terms of $u^{-}$. Since $u_{2}^{-}\leq c$ on $T$, by the smoothness of $\mathcal{W}_{\hskip 1.0ptc}^{-}$ at the origin (see Proposition 3.2), we get $u^{-}=\left(0,-1,0\right)$ at the origin. Since $u^{-}=\left(0,1,0\right)$ when $\rho=0$ and $z=1$, we can find a location on $l_{3}$ at where $u^{-}$ is singular. Denote by $x_{0}^{+}=\left(0,z_{0}\right)$ the lowest singularity of $u^{-}$ on $l_{3}^{+}$. Utilizing Proposition 4.7, for any $\epsilon>0$ sufficiently small, we have a $\delta\left(\epsilon\right)>0$ suitably small so that

[TABLE]

Here we use $\left(r_{*},\psi,\theta\right)$ to denote the spherical coordinate system with respect to the center $x_{0}^{+}$. Without ambiguity, $u^{-}$ in (5.6) is also understood as a $3$-vector field depending on $r_{*}$ and $\psi$.

In what follows we define a dumbbell.

[TABLE]

Similarly the horizontal line $z=-z_{0}+\delta\left(\epsilon\right)-\epsilon_{1}$ also has two intersections with ${\partial}D_{\delta\left(\epsilon\right)}\left(x_{0}^{-}\right)$. We denote by $x_{1}^{-}$ the intersection point with positive $x_{1}$-coordinate, while $x_{2}^{-}$ denotes another intersection point which has negative $x_{1}$-coordinate. The contour $\mathscr{C}$ in the $(x_{1},z)$ - plane is then defined as follows. Firstly we start from $x_{1}^{+}$ and rotate counter-clockwisely along ${\partial}D_{\delta\left(\epsilon\right)}\left(x_{0}^{+}\right)$ to $x_{2}^{+}$. Then we connect $x_{2}^{+}$ and $x_{2}^{-}$ by the vertical segment between them. From $x_{2}^{-}$, we rotate counter-clockwisely along ${\partial}D_{\delta\left(\epsilon\right)}\left(x_{0}^{-}\right)$ to $x_{1}^{-}$. Finally we connect $x_{1}^{-}$ and $x_{1}^{+}$ by the vertical segment between them. The dumbbell, denoted by $\mathrm{D}_{\epsilon,\epsilon_{1}}$, is the region in $(x_{1},z)$ - plane which is enclosed by the contour $\mathscr{C}$.

On \Big{\{}\left(\hskip 0.5pt0,z\right):z\in\left(z_{0},z_{0}+\delta\left(\epsilon\right)\right)\cup\left(-z_{0}-\delta\left(\epsilon\right),-z_{0}\right)\Big{\}}, it holds $u^{-}=\left(\hskip 0.5pt0,1,0\hskip 0.5pt\right)$. By Definition 1.3, $u^{-}$ is uniaxial with two eigenvalues equaling to -\sqrt{3}\big{/}6. The largest eigenvalue is \sqrt{3}\big{/}3, whose associated eigenspace is given by \mathrm{span}\Big{\{}e_{z}\Big{\}}. On \Big{\{}\left(\hskip 0.5pt0,z\right):z\in\left(-z_{0},z_{0}\right)\Big{\}}, it holds $u^{-}=\left(\hskip 0.5pt0,-1,0\hskip 0.5pt\right)$. Still by Definition 1.3, $u^{-}$ is uniaxial with two eigenvalues equaling to \sqrt{3}\big{/}6. The largest eigenvalue is \sqrt{3}\big{/}6, whose associated eigenspace is given by \mathrm{span}\Big{\{}e_{\theta},\hskip 1.0pte_{\rho}\Big{\}}.

In what follows we show the biaxiality of $u^{-}$ on $\mathrm{D}_{\epsilon,\epsilon_{1}}\sim l_{3}$. We firstly compare the three eigenvalues on $D_{\delta\left(\epsilon\right)}\left(x_{0}^{+}\right)$. Due to the symmetry of $u^{-}$ with respect to the variable $z$, the case for $D_{\delta\left(\epsilon\right)}\left(x_{0}^{-}\right)$ can be similarly studied. Suppose that $\sigma$ is an arbitrary number in \big{(}\hskip 1.0pt0,\delta\left(\epsilon\right)]. Using the polar angle $\psi$ in the spherical coordinate $\left(r_{*},\psi,\theta\right)$, we have

[TABLE]

Here we understand $u^{-}$ as a mapping depending on $r_{*}$ and $\psi$. In light of the regularity of $\mathcal{W}_{\hskip 1.0ptc}^{-}$ at $\left(\hskip 0.5pt0,z_{0}+\sigma\right)$, it follows {\partial}_{\psi}\hskip 0.5ptu^{-}\hskip 2.0pt\Big{|}_{\left(\sigma,0\right)}=0. The last equality is then reduced to

[TABLE]

which furthermore infers

[TABLE]

Applying the estimate in (5.6) to the last equality then yields

[TABLE]

Therefore we can fix $\epsilon$ sufficiently small and find a universal constant $\psi_{0}\in(\hskip 0.5pt0,\pi/4\hskip 0.5pt)$ so that

[TABLE]

Still by (5.6), on the sector with the polar angle \psi\in\big{[}\hskip 0.5pt\psi_{0},\pi\hskip 0.5pt\big{]}, it holds

[TABLE]

In light of the last estimate, we can keep choosing $\epsilon$ small enough and obtain \sqrt{3}\hskip 1.0ptu_{1}^{-}\hskip 1.0pt\Big{|}_{\left(\sigma,\psi\right)}+u_{2}^{-}\hskip 1.0pt\Big{|}_{\left(\sigma,\psi\right)}\hskip 2.0pt<\hskip 2.0pt1, for any \sigma\in\big{(}\hskip 1.0pt0,\delta\left(\epsilon\right)\big{]} and \psi\in\big{[}\hskip 0.5pt\psi_{0},\pi\hskip 0.5pt\big{]}. This estimate and (5.8) infer

[TABLE]

When $\psi\in(0,\pi)$, $u_{1}^{-}$ is strictly positive. It turns out

[TABLE]

Utilizing the last two bounds then yields

[TABLE]

Equivalently we have

[TABLE]

By Definition 1.3, it holds

[TABLE]

Moreover we also have

[TABLE]

Applying (5.10) to the right-hand sides of (5.11)-(5.12) and utilizing the symmetry of $u^{-}$ with respect to the variable $z$, we get

[TABLE]

Now we fix a sufficiently small $\epsilon$ so that (5.13) holds. We are left to compare the three eigenvalues on $\mathrm{R}_{\star}$, where $\mathrm{R}_{\star}$ is the rectangle in $(x_{1},z)$ - plane with four vertices given by $x_{1}^{+}$, $x_{1}^{-}$, $x_{2}^{+}$, $x_{2}^{-}$. Since $u^{-}$ is regular on $\mathrm{R}_{\star}$, $u^{-}$ is close to $\left(\hskip 1.0pt0,-1,0\hskip 0.5pt\right)$ on $\mathrm{R}_{\star}$, provided $\epsilon_{1}$ is small enough. It turns out $\lambda_{1}>\lambda_{2}$ on $\mathrm{R}_{\star}$. Using the strict positivity of $u_{1}^{-}$, we have

[TABLE]

Moreover $\sqrt{3}\hskip 1.0ptu_{1}^{-}+u_{2}^{-}<1$ on $\mathrm{R}_{\star}$ in that $u^{-}$ is close to $\left(\hskip 1.0pt0,-1,0\hskip 0.5pt\right)$ on $\mathrm{R}_{\star}$. It turns out $\left(\sqrt{3}\hskip 1.0ptu_{1}^{-}+u_{2}^{-}\right)^{2}\hskip 2.0pt<\hskip 2.0pt1$ on $\mathrm{R}_{\star}\sim l_{3}$. Equivalently we have $\left(u_{1}^{-}-\sqrt{3}\hskip 1.0ptu_{2}^{-}\right)^{2}+4\left(u_{3}^{-}\right)^{2}\hskip 2.0pt>\hskip 2.0pt9\left(u_{1}^{-}+\dfrac{1}{\sqrt{3}}\hskip 1.0ptu_{2}^{-}\right)^{2}$ on $\mathrm{R}_{\star}\sim l_{3}$. Therefore $\lambda_{3}>\lambda_{1}$ on $\mathrm{R}_{\star}\sim l_{3}$ by this inequality and (5.12). Hence it holds $\lambda_{3}\hskip 1.0pt>\hskip 1.0pt\lambda_{1}\hskip 1.0pt>\hskip 1.0pt\lambda_{2}$ on $\mathrm{R}_{\star}\sim l_{3}$, which together with (5.13) then infer the biaxiality of $u^{-}$ on $\mathrm{D}_{\epsilon,\epsilon_{1}}\sim l_{3}$. More precisely we have

[TABLE]

In the end we describe the variation of director field near the vertical segment connecting $x_{0}^{+}$ and $x_{0}^{-}$. In light of (5.14), the largest eigenvalue is $\lambda_{3}$ on $\mathrm{D}_{\epsilon,\epsilon_{1}}\sim l_{3}$. Therefore $\kappa_{\star}$ in (4) of Theorem 1.7 is a director field determined by $u^{-}$ on $\mathrm{D}_{\epsilon,\epsilon_{1}}\sim l_{3}$. Fixing \big{(}0,0,z\big{)} with z\in\big{(}\hskip 0.5ptz_{0},\hskip 1.0ptz_{0}+\delta\left(\epsilon\right)\hskip 1.0pt\big{]}, we have $u^{-}$ approaching to $(0,1,0)$ while $x$ approaches to $(0,0,z)$. It then turns out

[TABLE]

Since $u_{3}^{-}$ is positive when $x$ is close to $(0,0,z)$, we have

[TABLE]

which infers

[TABLE]

Applying this limit and (5.15), we obtain

[TABLE]

Utilizing the strict negativity of $u_{3}^{-}$ in the lower-half plane, we also have

[TABLE]

As for $(0,0,z)$ with $z\in(-z_{0},z_{0})$, $u^{-}$ is close to $(0,-1,0)$ when $x$ is close to $(0,0,z)$. It follows

[TABLE]

Moreover by $u_{3}^{-}=0$ on $T$, we also have

[TABLE]

Now we denote by $\mathscr{C}^{+}$ the part of $\mathscr{C}$ with positive $x_{1}$ - coordinate. By (5.16)-(5.19), the director field $\kappa_{\star}$ varies from $-e_{z}$ to $e_{\rho}$ and then to $e_{z}$ when points vary from \big{(}0,0,-z_{0}-\delta\left(\epsilon\right)\big{)} to \big{(}0,0,z_{0}+\delta\left(\epsilon\right)\big{)} along the contour $\mathscr{C}^{+}$. Since $\kappa_{\star}$ has positive coefficient in front of $e_{\rho}$ for all points on $\mathscr{C}^{+}\sim l_{3}$, therefore $\kappa_{\star}$ keeps on the right-half part of $(\rho,z)$-plane for all points on $\mathscr{C}^{+}\sim l_{3}$. When points are changed from \big{(}0,0,-z_{0}-\delta\left(\epsilon\right)\big{)} to \big{(}0,0,z_{0}+\delta\left(\epsilon\right)\big{)} along $\mathscr{C}^{+}$, the angle of $\kappa_{\star}$ is totally changed by $\pi$. Part (5) in Theorem 1.7 follows from (5.6), (5.18) and the representation of $\kappa_{\star}$.

**VI. CONVERGENCE RESULTS

**In this last section, we prove Theorem 1.8 and the convergence result in Theorem 1.5.

**VI.1. PROOF OF THEOREM 1.8

$\mathcal{W}_{b}^{+}$** is uniformly bounded in $H^{1}(B_{1};\mathbb{S}^{4})$ for all $b\in\mathrm{I}_{-}$. Up to a subsequence, there is a $\mathcal{W}_{\star}^{+}\in H^{1}(B_{1};\mathbb{S}^{4})$ so that $\mathcal{W}_{b}^{+}$ converges weakly in $H^{1}(B_{1};\mathbb{S}^{4})$ to $\mathcal{W}_{\star}^{+}$ as $b\rightarrow-1$. $\mathcal{W}_{\star}^{+}$ is a weak solution to (1.13). Moreover there is a $\mathbb{S}^{2}$-valued mapping $u_{\star}^{+}$ on $B_{1}$ so that \mathcal{W}_{\star}^{+}=L\hskip 1.0pt\big{[}\hskip 1.0ptu_{\star}^{+}\hskip 1.0pt\big{]}. Since the energy E_{\mu}\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]} is non-decreasing with respect to $b$, it holds by lower semi-continuity that

[TABLE]

Fix $x_{0}\in l_{3}^{+}$ and a radius $r$ satisfying $B_{r}(x_{0})\subset\subset B_{1}^{+}$. By the fact that $u_{b}^{+}$ is the minimizer of the first minimization problem in (1.12), we can apply similar construction as in the proof of convergence theorem 5.5 in [31] to show $\mathcal{W}_{b}^{+}$ converges to $\mathcal{W}_{\star}^{+}$ strongly in $H^{1}(B_{r}(x_{0});\mathbb{S}^{4})$. As for points on $\mathbb{D}$, there is no obstruction to prevent $u_{b}^{+}$ converging to $u_{\star}^{+}$ strongly in $H^{1}_{\mathrm{loc}}(\mathbb{D};\mathbb{S}^{2})$. Otherwise by standard bubbling analysis (see [51]), there will be a non-trivial harmonic map from $\mathbb{R}^{2}$ to $\mathbb{S}^{2}$ which is a bubble map obtained by this bubbling analysis. However this is impossible since for all $b\in\mathrm{I}_{-}$, $u_{b;1}^{+}>0$ on $\mathbb{D}$. At the origin, by (3.1) and (1.12), it holds

[TABLE]

Hence there is a universal constant $c>0$ so that for any $r_{0}\in(0,1)$ and $\sigma\in(0,r_{0})$,

[TABLE]

Here we have used (6.2) and (3.1) to get the last inequality above. By taking $b\rightarrow-1$ and $\sigma\rightarrow 0$ successively in the last estimate, it turns out $\mathcal{W}_{b}^{+}$ converges to $\mathcal{W}_{\star}^{+}$ strongly in $H^{1}(B_{r_{0}};\mathbb{S}^{4})$. In summary we have $\mathcal{W}_{b}^{+}$ converges to $\mathcal{W}_{\star}^{+}$ strongly in $H^{1}_{\mathrm{loc}}(B_{1};\mathbb{S}^{4})$. Applying this strong convergence and Fatou’s lemma to (3.1), we also have (3.1) holds for $\mathcal{W}_{\star}^{+}$. Similar monotonicity inequality holds for $\mathcal{W}_{\star}^{+}$ at other points in $B_{1}$. Therefore (6.1) and proof of Proposition 3.2 can be applied to infer the smoothness of $\mathcal{W}_{\star}^{+}$ at the origin. More precisely $\mathcal{W}_{\star}^{+}$ is smooth in a neighborhood of [math]. By this smoothness result and $H^{1}_{\mathrm{loc}}(B_{1};\mathbb{S}^{4})$-convergence of $\mathcal{W}_{b}^{+}$ as $b\rightarrow-1$, for $\epsilon>0$ sufficiently small, we can find a uniform radius $r_{\star}$ and $b_{1}\in\mathrm{I}_{-}$ so that \displaystyle r_{\star}^{-1}\int_{B_{r_{\star}}}\big{|}\hskip 1.0pt\nabla\mathcal{W}_{b}^{+}\hskip 1.0pt\big{|}^{2}<\epsilon, for all $b\in(-1,b_{1})$. By standard $\epsilon$-regularity result and Arzelà-Ascoli theorem, $\mathcal{W}_{b}^{+}$ converges to $\mathcal{W}_{\star}^{+}$ uniformly in a neighborhood of [math]. Therefore $\mathcal{W}_{\star}^{+}$ is continuous on $T$ with its third component $\mathcal{W}_{\star;3}^{+}=1$ at [math]. Utilizing maximum principle we also have $u_{\star;1}^{+}>0$ on $\mathbb{D}$. There then exists a $b_{\star}\in\mathrm{I}_{-}$ so that $u_{\star;2}^{+}=\mathcal{W}_{\star;3}^{+}\geq b_{\star}$ on $T$. That is $u_{\star}^{+}\in\mathscr{F}_{b_{\star},+}$, which induces E_{\mu}\big{[}\hskip 1.0ptu_{\star}^{+}\hskip 1.0pt\big{]}\geq\mathrm{Min}\hskip 2.0pt\bigg{\{}\hskip 1.0ptE_{\mu}\left[\hskip 1.0ptu\right]:u\in\mathscr{F}_{\hskip 1.0ptb_{\star},+}\bigg{\}}. By this lower bound and (6.1), $u_{\star}^{+}$ is a minimizer of $E_{\mu}$-energy on $\mathscr{F}_{\hskip 1.0ptb_{\star},+}$. $u_{\star}^{+}$ is then the $u_{b_{\star}}^{+}$ in Theorem 1.8. Moreover by lower semi-continuity (or (6.1)), it holds \displaystyle E_{\mu}\big{[}\hskip 1.0ptu_{\star}^{+}\hskip 1.0pt\big{]}\hskip 2.0pt\leq\hskip 2.0pt\liminf_{b\hskip 1.0pt\rightarrow\hskip 1.0pt-1}E_{\mu}\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]}. Utilizling the non-decreasing monotonicity of E_{\mu}\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]} yields \displaystyle\limsup_{b\hskip 1.0pt\rightarrow\hskip 1.0pt-1}E_{\mu}\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]}\hskip 2.0pt\leq\hskip 2.0ptE_{\mu}\big{[}\hskip 1.0ptu_{\star}^{+}\hskip 1.0pt\big{]}. Then we induce that E_{\mu}\big{[}\hskip 1.0ptu_{b}^{+}\hskip 1.0pt\big{]} converges to E_{\mu}\big{[}\hskip 1.0ptu_{\star}^{+}\hskip 1.0pt\big{]} as $b\rightarrow-1$.

Denote by $u_{\star}$ a minimizer of $E_{\mu}$-energy on $\mathscr{F}^{s}$. Moreover we assume $u_{\star;1}\geq 0$, $u_{\star;3}\geq 0$ on $B_{1}^{+}$. Similar arguments for $u_{b}^{+}$ or $u_{c}^{-}$ can be applied to show that $u_{\star}$ is regular on $T$. Let $f$ be a smooth radial function compactly supported in $B_{1}$ and equal to $1$ in $B_{1/2}$. Utilizing $f$ and $U^{*}$, we construct $u_{\star}+\epsilon fU^{*}$ and denote by $u_{\star}^{\epsilon}$ the normalized vector field of $u_{\star}+\epsilon fU^{*}$. There exists a $b_{\epsilon}\in\mathrm{I}_{-}$ satisfying $b_{\epsilon}\rightarrow-1$ as $\epsilon\rightarrow 0$ so that $u_{\star;2}^{\epsilon}\geq b_{\epsilon}$ on $T$ in the sense of trace. It then holds E_{\mu}\big{[}\hskip 1.0ptu_{\star}^{\epsilon}\hskip 1.0pt\big{]}\hskip 2.0pt\geq\hskip 2.0ptE_{\mu}\big{[}\hskip 1.0ptu_{b_{\epsilon}}^{+}\hskip 1.0pt\big{]}\hskip 2.0pt=\hskip 2.0ptE_{\mu}\big{[}\hskip 1.0ptu_{\star}^{+}\hskip 1.0pt\big{]}. Taking $\epsilon\rightarrow 0$ yields E_{\mu}\big{[}\hskip 1.0ptu_{\star}\hskip 1.0pt\big{]}\geq E_{\mu}\big{[}\hskip 1.0ptu_{\star}^{+}\hskip 1.0pt\big{]}, which furthermore infers E_{\mu}\big{[}\hskip 1.0ptu_{\star}\hskip 1.0pt\big{]}=E_{\mu}\big{[}\hskip 1.0ptu_{\star}^{+}\hskip 1.0pt\big{]}. The case when $c\rightarrow 1$ can be similar discussed. The proof is then completed.

**VI.2. PROOF OF CONVERGENCE RESULT IN THEOREM 1.5

**Suppose $u_{\mu}$ is a minimizer of $E_{\mu}$-energy within configuration space $\mathscr{F}^{s}$. It can be shown that up to a subsequence, denoted by $\mu_{k}$, $u_{\mu_{k}}$ converges strongly in $H^{1}(B_{1})$ to a limit $u^{\star}_{\infty}$ as $k\rightarrow\infty$. Moreover

Let $u^{\star}_{\infty;j}$ be the $j$-th component of $u^{\star}_{\infty}$. On $B_{1}$, we define $u^{\dagger}_{1}:=|\hskip 1.0ptu^{\star}_{\infty;1}\hskip 1.0pt|$, $u^{\dagger}_{2}:=u^{\star}_{\infty;2}$, $u^{\dagger}_{3}:=u^{\star}_{\infty;3}$. It holds

We can also characterize $u^{\dagger}$ in terms of a minimization problem on $2$-vector fields.

[TABLE]

Here ${\partial}^{+}B_{1}$ denotes the spherical boundary of $B_{1}^{+}$. Then it holds

[TABLE]

Moreover $v^{\dagger}$ is a minimizer of the energy $F$ on $\mathscr{F}_{2}$.

[TABLE]

Therefore it holds

[TABLE]

Otherwise from (6.5), we have $\sqrt{3}\hskip 1.5ptu^{\dagger}_{1}+u^{\dagger}_{2}=-2$ on a subset of $B^{+}_{1}$ with positive Lebesgue measure. Since \big{(}u^{\dagger}_{1}\hskip 0.5pt\big{)}^{2}+\big{(}u^{\dagger}_{2}\hskip 0.5pt\big{)}^{2}\leq 1, it then follows $u^{\dagger}_{1}=-\sqrt{3}\hskip 1.0pt/\hskip 1.0pt2$ and $u^{\dagger}_{2}=-1\hskip 1.0pt/\hskip 1.0pt2$ on this subset. This is a contradiction to the almost non-negativity of $u^{\dagger}_{1}$ on $B^{+}_{1}$. Plugging $u^{\dagger}_{2}=1-\sqrt{3}\hskip 1.5ptu^{\dagger}_{1}$ into $\left|\hskip 1.0ptu^{\dagger}\hskip 1.0pt\right|^{2}=1$, we obtain

[TABLE]

In light of $v_{1}^{\dagger}$ and $v_{2}^{\dagger}$ given in (6.4), by (6.7) $v^{\dagger}$ must have unit length for almost all points on $B_{1}^{+}$. This result combined with $u^{\dagger}\in\mathscr{F}_{1}$ induces $v^{\dagger}\in\mathscr{F}_{2}$. In terms of $v^{\dagger}$, the energy E\hskip 1.0pt\big{[}\hskip 1.0ptu^{\dagger}\hskip 1.0pt\big{]} (see (1.8)) can be evaluated by

[TABLE]

Since $v^{\dagger}\in\mathscr{F}_{2}$, the above equality infers

[TABLE]

Letting $v=(v_{1},v_{2})\in\mathscr{F}_{2}$ be arbitrarily given, we define u^{v}_{1}:=\sqrt{3}\big{(}1-v_{1}\big{)}\big{/}4, u^{v}_{2}:=\big{(}1+3v_{1}\big{)}\big{/}4 and u^{v}_{3}:=\sqrt{3}v_{2}\big{/}2. Moreover we extend $u^{v}:=\left(u^{v}_{1},u^{v}_{2},u^{v}_{3}\right)$ to $B_{1}$ so that the extension, still denoted by $u^{v}$, is $\mathscr{R}$-axially symmetric. $u^{v}\hskip 1.0pt\in\hskip 1.0pt\mathscr{F}_{1}$. By Lemma 6.2, it holds E\big{[}\hskip 1.0ptu^{\dagger}\hskip 1.0pt\big{]}\hskip 2.0pt\leq\hskip 2.0ptE\left[\hskip 1.0ptu^{v}\hskip 1.0pt\right]\hskip 2.0pt=\hskip 2.0pt\dfrac{3}{2}F\left[\hskip 0.5ptv\hskip 0.5pt\right]. Taking minimum over $v\in\mathscr{F}_{2}$, we obtain from this estimate that E\big{[}\hskip 1.0ptu^{\dagger}\hskip 1.0pt\big{]}\hskip 2.0pt\leq\hskip 2.0pt\dfrac{3}{2}\hskip 2.0pt\mathrm{Min}\hskip 2.0pt\Big{\{}F\left[\hskip 0.5ptv\hskip 0.5pt\right]:v\in\mathscr{F}_{2}\hskip 1.0pt\Big{\}}. The lemma then follows by this upper bound and (6.8)-(6.9). ∎

By regularity result in [49], we can obtain the regularity of $v^{\dagger}$, which infers the following regularity of $u^{\dagger}$.

Here u^{\dagger}\equiv\big{(}\sqrt{3}\big{/}2,\hskip 1.0pt-1/2,\hskip 1.0pt0\hskip 1.0pt\big{)} on  $T^{\circ}$ follows from (6.6), $u_{3}^{\dagger}=0$ on $T$, the unit length of $u^{\dagger}$, the continuity of $u^{\dagger}$ on T\sim\big{\{}0\big{\}} and the fact that u^{\dagger}(1,0)=\big{(}\sqrt{3}\big{/}2,\hskip 1.0pt-1/2,\hskip 1.0pt0\hskip 1.0pt\big{)}. Now we show

Step 1. Lower Bound. Suppose that $\beta=\beta(\rho,z)$ is an angular function on $B_{1}$ satisfying

[TABLE]

Associated with $\beta$, we define w_{\beta}:=\big{(}\sin\beta\cos\theta,\sin\beta\sin\theta,\cos\beta\big{)}. The map $w_{\beta}$ is an $\mathbb{S}^{2}$-valued vector field on $B_{1}$. It satisfies $\displaystyle\int_{B_{1}}|\hskip 1.0pt\nabla w_{\beta}\hskip 1.0pt|^{2}=\int_{B_{1}}|\hskip 0.5pt\nabla\beta\hskip 0.5pt|^{2}+\frac{1}{\rho^{2}}\sin^{2}\beta\hskip 2.0pt<\hskip 2.0pt\infty$. Moreover $w_{\beta}(x)=x$ on ${\partial}B_{1}$. Using Theorem 7.1 in [11], we have

[TABLE]

Step 2. Lifting and Extension. On $B_{1}^{+}$, we define $u^{\ddagger}_{1}:=u^{\dagger}_{1}$, $u^{\ddagger}_{2}:=u^{\dagger}_{2}$, u^{\ddagger}_{3}:=\big{|}\hskip 2.0ptu^{\dagger}_{3}\hskip 2.0pt\big{|}. By (6.7) and the almost non-negativity of $u^{\ddagger}_{3}$ on $B_{1}^{+}$, there exists a unique angular function $\alpha_{+}$ with values in $[\hskip 0.5pt0,\pi\hskip 0.5pt/\hskip 0.5pt2\hskip 1.0pt]$ so that

[TABLE]

Moreover \alpha_{+}\in H^{1}\big{(}B_{1}^{+};[\hskip 1.0pt0,\pi\hskip 1.0pt/\hskip 1.0pt2]\hskip 1.0pt\big{)}. For all $(x_{1},x_{2},z)\in B_{1}^{-}$, we define $\alpha_{-}(x_{1},x_{2},z)=\pi-\alpha_{+}(x_{1},x_{2},-z)$. With $\alpha_{+}$ and $\alpha_{-}$, we introduce the angular function $\alpha$ on $B_{1}$ which equals to $\alpha_{+}$ and $\alpha_{-}$ on $B_{1}^{+}$ and $B_{1}^{-}$, respectively. By Lemma 6.4 and (6.12), it holds $\cos 2\alpha_{+}=-1$ on T\sim\big{\{}\hskip 0.5pt0\hskip 0.5pt\big{\}}. Since $\alpha_{+}\in[\hskip 1.0pt0,\pi\hskip 1.0pt/2\hskip 1.0pt]$, we have \alpha_{+}=\pi\hskip 1.0pt\big{/}\hskip 1.0pt2 on T\sim\big{\{}\hskip 0.5pt0\hskip 0.5pt\big{\}}. Therefore the trace of \alpha_{+}-\pi\hskip 1.0pt\big{/}\hskip 1.0pt2 equals to [math] on $T$. Note that \alpha-\pi\hskip 1.0pt\big{/}\hskip 1.0pt2 is the odd extension of \alpha_{+}-\pi\hskip 1.0pt\big{/}\hskip 1.0pt2 from $B_{1}^{+}$ to $B_{1}$. It then turns out \alpha-\pi\hskip 1.0pt\big{/}\hskip 1.0pt2\in H^{1}\big{(}B_{1}\big{)} by the $H^{1}$-regularity of $\alpha_{+}$ on $B_{1}^{+}$. Equivalently we have the $H^{1}$-regularity of $\alpha$ on $B_{1}$. Utilizing $u^{\ddagger}=U^{*}$ on ${\partial}^{+}B_{1}$ and noticing the range of $\alpha_{+}$, we get $\alpha_{+}=\varphi$ on ${\partial}^{+}B_{1}$, which in turn induces $\alpha=\varphi$ on ${\partial}B_{1}$.

Step 3. Upper Bound and Completion of Proof. The angular function $\alpha$ constructed in Step 2 satisfies (6.10). Then we have a $\mathbb{S}^{2}$-valued map $w_{\alpha}=(\sin\alpha\cos\theta,\sin\alpha\sin\theta,\cos\alpha)$. The Dirichlet energy of $w_{\alpha}$ can be calculated by

[TABLE]

In light of (6.12), it holds

[TABLE]

Now we extend $u^{\ddagger}:=\left(u^{\ddagger}_{1},u^{\ddagger}_{2},u^{\ddagger}_{3}\right)$ to $B_{1}$ so that the extension, still denoted by $u^{\ddagger}$, is $\mathscr{R}$-axially symmetric. Then by (6.8), the last equality can be reduced to $\displaystyle\int_{B_{1}^{+}}|\hskip 0.5pt\nabla\alpha_{+}\hskip 0.5pt|^{2}+\frac{1}{\rho^{2}}\sin^{2}\alpha_{+}=\dfrac{1}{6}\hskip 1.0ptE\left[\hskip 1.0ptu^{\ddagger}\hskip 1.0pt\right]$. Utilizing Lemma 6.2 and the fact $U^{*}\in\mathscr{F}_{1}$, we have E\big{[}\hskip 0.5ptu^{\ddagger}\hskip 0.5pt\big{]}\leq E\big{[}\hskip 0.5ptu^{\dagger}\hskip 0.5pt\big{]}\hskip 2.0pt\leq\hskip 2.0ptE\big{[}\hskip 1.0ptU^{*}\hskip 0.5pt\big{]}\hskip 2.0pt=\hskip 2.0pt24\pi. The above arguments infer \displaystyle\int_{B_{1}}\big{|}\hskip 1.0pt\nabla w_{\alpha}\hskip 1.0pt\big{|}^{2}\leq 8\hskip 1.0pt\pi. Using this upper bound and (6.11) then yields \displaystyle\int_{B_{1}}\big{|}\hskip 1.0pt\nabla w_{\alpha}\hskip 1.0pt\big{|}^{2}=8\hskip 1.0pt\pi. Equivalently $w_{\alpha}$ saturates the minimum Dirichlet energy within the space \bigg{\{}u\in H^{1}\big{(}B_{1};\mathbb{S}^{2}\big{)}:u(x)=x\hskip 5.0pt\text{on {\partial}B_{1}}\bigg{\}}. Applying Theorem 7.1 in [11] then yields w_{\alpha}=x\hskip 1.0pt\big{/}\hskip 1.0pt|\hskip 0.5ptx\hskip 0.5pt| on $B_{1}$. That is $\alpha=\varphi$ on $B_{1}$. This result show that $u^{\ddagger}=U^{*}$ on $B_{1}$. Using $P(u^{\star}_{\infty})=1/3$ and the regularity result in Lemma 6.4, we then obtain $u^{\star}_{\infty}=U^{*}$ on $B_{1}$. The proof is finished. ∎

ACKNOWLEDGMENTS The author is partially supported by RGC grant of Hong Kong No. 14306414.

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YONG YU

Department of Mathematics

The Chinese University of Hong Kong

E-mail: yongyu@math.cuhk.edu.hk