\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 34, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/34\hfil Regularity of harmonic maps] {Another proof of the regularity of harmonic maps from a Riemannian manifold to \\ the unit sphere} \author[J. Aramaki \hfil EJDE-2014/34\hfilneg] {Junichi Aramaki} % in alphabetical order \address{Junichi Aramaki \newline Division of Science, Faculty of Science and Engineering, Tokyo Denki University, \newline Hatoyama-machi, Saitama 350-0394, Japan} \email{aramaki@mail.dendai.ac.jp} \thanks{Submitted September 16, 2013. Published January 27, 2014.} \subjclass[2000]{58E20, 53C43, 58E30} \keywords{Harmonic maps; minimizing harmonic maps; weak Harnack inequality} \begin{abstract} We shall consider harmonic maps from $n$-dimensional compact connected Riemannian manifold with boundary to the unit sphere under the Dirichlet boundary condition. We claim that if the Dirichlet data is smooth and so-called ``small'', all minimizers of the energy functional are also smooth and ``small''. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Let $(M,g)$ be a $n$-dimensional Riemannian manifold with boundary $\partial M$ endowed with a smooth Riemannian metric $g$. For any $p \in M$, let $(x_1,\ldots ,x_n)$ be a coordinate system near $p$. Then $g$ can be represented by \[ g= \sum _{\alpha ,\beta =1}^n g_{\alpha \beta }dx_{\alpha }\otimes dx_{\beta } \] where $(g_{\alpha \beta })$ is a positive definite symmetric $n\times n$ matrix. We write the inverse matrix of $(g_{\alpha \beta })$ by $(g^{\alpha \beta })$ and the volume element of $(M,g)$ by $dv_g= \sqrt{g}dx$ where $g= \det (g_{\alpha \beta })$, and we use the notations that for any vector fields $\mathbf{u} ,\mathbf{v} $, $\langle \mathbf{u} ,\mathbf{v} \rangle _g= g(\mathbf{u} ,\mathbf{v} )$ and $|\mathbf{u} |^2 _g= \langle \mathbf{u} ,\mathbf{u} \rangle _g$. We view maps from $M$ into a $k$-dimensional unit sphere $\mathbb{S}^k \subset \mathbb{R} ^{k+1}$, extrinsically. The Sobolev space $W^{1,2}(M, \mathbb{R} ^{k+1})$ is standardly defined and the space $W^{1,2}(M, \mathbb{S}^k)$ is defined by \[ W^{1,2}(M, \mathbb{S}^k )=\big\{\mathbf{u} = (u^1,\ldots ,u^{k+1}) \in W^{1,2}(M,\mathbb{R} ^{k+1}); \; \mathbf{u} (x) \in \mathbb{S}^k \text{ a.e. } x \in M\big\}. \] For any $\mathbf{u} \in W^{1,2}(M,\mathbb{S}^k)$, the Dirichlet energy density is defined by \begin{equation} e(\mathbf{u} )= \frac12 |\nabla \mathbf{u} |_g ^2 \label{e1.1} \end{equation} where $|\nabla \mathbf{u} |_g ^2 = \sum _{i=1}^{k+1} |\nabla u^i|_g^2$. In any local coordinate system $x=(x_1, \ldots , x_n)$, we see that \[ e(\mathbf{u} )= \frac12 \sum _{\alpha ,\beta =1}^n \sum _{i=1}^{k+1} g^{\alpha \beta } \frac{\partial u^i}{\partial x_{\alpha }} \frac{\partial u^i}{\partial x_{\beta }}, \] and the Dirichlet energy is defined by \begin{equation} E(\mathbf{u} ,M)= \int _M e(\mathbf{u} )dv_g. \label{e1.2} \end{equation} We say $\mathbf{u} \in W^{1,2}(M,\mathbb{S}^k)$ is weakly harmonic map, if \begin{equation} \int _M \sum _{\alpha ,\beta =1}^n g^{\alpha \beta } \Big( \frac{\partial \mathbf{u} }{\partial x_{\alpha }} \cdot \frac{\partial \boldsymbol{\phi} }{\partial x _{\beta }} + \Bigl( \frac{\partial \mathbf{u} }{\partial x_{\alpha }} \cdot \frac{\partial \mathbf{u} }{\partial x_{\beta }}\Bigr) \mathbf{u} \cdot \boldsymbol{\phi} \Big)dv_g=0 \label{e1.3} \end{equation} for any $\boldsymbol{\phi} \in C_0^{\infty }(M, \mathbb{R} ^{k+1})$ where $\cdot $ denotes the Euclidean inner product in $\mathbb{R} ^{k+1}$. Then $\mathbf{u} $ satisfies the harmonic map equation in the sense of distribution \begin{equation} \Delta _g \mathbf{u} + \sum _{\alpha ,\beta =1}^n g^{\alpha \beta } \frac{\partial \mathbf{u} }{\partial x_{\alpha }}\cdot \frac{\partial \mathbf{u} }{\partial x_{\beta }} \mathbf{u} =\mathbf{0}\quad \text{in } M \label{e1.4} \end{equation} where $\Delta _g$ is the Laplace-Beltrami operator on $(M,g)$ given by \[ \Delta _g = \frac{1}{\sqrt{g}} \sum _{\alpha ,\beta =1}^n \frac{\partial }{\partial x_{\alpha }} \Big( \sqrt{g} g^{\alpha \beta } \frac{\partial }{\partial x_{\beta }}\Big). \] Next we say $\mathbf{u} \in W^{1,2}(M,\mathbb{S}^k)$ is a minimizing harmonic map, if for any $\Omega \subset M$, \begin{equation} E(\mathbf{u} ,\Omega ):=\int _{\Omega } e(\mathbf{u} ) dv_g \le E(\mathbf{v} ,\Omega ) \label{e1.5} \end{equation} for all $\mathbf{v} \in W^{1,2}(\Omega ,\mathbb{S}^k)$ with $\mathbf{v} |_{\partial M }= \mathbf{u} |_{\partial M }$. The regularity of minimizing harmonic maps has been studied by many authors for a general target Riemannian manifold $N$ instead of $\mathbb{S}^k$. For the case where $\dim M=2$, Morrey \cite{Mo} showed that if $\mathbf{u} \in W^{1,2}(M,N)$ is a minimizing harmonic map, then $\mathbf{u} \in C^{\infty }(M,N)$. For $n\ge 3$, Schoen and Uhlenbeck \cite{SU82} have shown that if we define the singular set of any minimizing map $\mathbf{u} \in W^{1,2}(M,N)$ by \[ \operatorname{sing}(\mathbf{u} )=\{x \in M; \mathbf{u} \text{ is discontinuous at } x\}, \] then $\operatorname{sing}(\mathbf{u} )$ is a closed set, and it is discrete for $n=3$, and \[ \dim _H(\operatorname{sing}(\mathbf{u} ))\le n-3 \] for $n\ge 4$ where $\dim _H(\operatorname{sing}(\mathbf{u} ))$ is the Hausdorff dimension of $\operatorname{sing}(\mathbf{u} )$. Moreover, it is well known that $\mathbf{u} $ is analytic in $M\setminus \operatorname{sing}(\mathbf{u} )$ (cf. Borchers and Garber \cite{BG}). For $p \in N, r>0$, let $B_r(p)=\{q \in N; \mathrm{dist} _N(q,p)\le r\}$ be the closed geodesic ball with center $p$ and radius $r$, and let $C(p)$ be the cut locus of $p$. We call $B_r(p)$ is a regular ball if the following two conditions hold. \begin{itemize} \item[(i)] $\sqrt{\kappa } r < \pi /2$ where $\kappa = \max \{ 0, \sup _{B_r(p)}K^N\}$, $K^N$ is the sectional curvature of $N$. \item[(ii)] $C(p) \cap B_r(p)= \emptyset $. \end{itemize} Hildebrandt et al. \cite{HKW} have established the following existence theorem of smooth harmonic maps with given boundary data contained in a regular ball. (see also Lin and Wang \cite[Theorem 3.1.7]{linwang}). \begin{theorem}[\cite{HKW}] \label{thm1.1} Suppose that $B_r(p)\subset N$ is a regular ball and $\Omega \subset M$ is a bounded domain and $\mathbf{g} :\Omega \to B_r(p)$ is continuous map and has finite energy. Then there exists a harmonic map $\mathbf{u} \in C^{2+\alpha }(\Omega ,N)\cap C^0(\overline{\Omega },N)$ with $\mathbf{u} |_{\partial M }=\mathbf{g} $. \end{theorem} As the first step of their proof, they considered the following variational problem. Find a minimizer of \[ \inf _{\mathbf{u} \in V}\int _{\Omega } e(\mathbf{u} ) dv_g \] where the admissible space $V$ is as follows. Choose $r_1\in (r, \pi /2\sqrt{\kappa })$ such that $B_{r_1}(p)\subset N$ is also regular ball, and define \[ V=\{\mathbf{u} \in W^{1,2}(\Omega ,B_{r_1}(p)); \mathbf{u} |_{\partial M } = \mathbf{g} \}. \] This admissible space seems to be restrictive. Thus in the present paper, we report that in order to get the same result for the target manifold $N=\mathbb{S}^k$, we can take the admissible space $V=W^{1,2}(M ,\mathbb{S}^k,\mathbf{g} ):=\{\mathbf{u} \in W^{1,2}(M ,\mathbb{S}^k); \mathbf{u} |_{\partial M }=\mathbf{g} \}$. We note that in the case where $N=\mathbb{S}^{k}$, since $K^N=1$ and $C(p)=\{-p\}$, if $00$, we denote the closed geodesic ball in $\mathbb{S}^k$ with center $p$ and radius $r$ by $B_r(p)$. Throughout this paper we treat the $B_r(p)$ which is an closed ball with $00$ on $\partial M $, we can see that $\mathbf{w} \in W^{1,2}(M ,\mathbb{S}^k ,\mathbf{e} )$, and $\mathbf{w} $ is also a minimizer of \eqref{e2.3}. Therefore $\mathbf{w} $ also satisfies \eqref{e3.2}, and $\mathbf{w} \in C^{2+\alpha }$ near the boundary (cf. Schoen and Uhlenbeck \cite[Proposition 3.1]{SU83}. In particular, $w^1$ satisfies $w^1\ge 0$ and \begin{equation} \begin{gathered} -\Delta _g w^1 = |\nabla \mathbf{w} |^2_g w^1 \quad \text{in } M ,\\ w^1 = e^1 \quad \text{on } \partial M \end{gathered} \label{e3.3} \end{equation} For any $q \in M$, choose a local coordinate neighborhood $U_q$ and a local coordinate system $(x_1, \ldots ,x_n)$. Then $w^1$ is a bounded non-negative weak supersolution of \[ \Delta _g= \frac{1}{\sqrt{g}}\sum _{\alpha ,\beta =1}^n \frac{\partial }{\partial x_{\alpha }} \Big( \sqrt{g}g^{\alpha \beta }\frac{\partial }{\partial x_{\beta }}\Big); \] that is to say, $\Delta _g w^1\le 0$ in $U_q $. We can apply the weak Harnack inequality (cf. \cite[Theorem 8.18]{GilTru} or \cite[Chapter 4, Lemma 1.3]{CW}). Thus for any $1\le p < n/(n-2)$, $B_{2R} \subset U_q $ \[ \operatorname{ess\,inf} _{B_{ R}} w^1 \ge c\Big( \frac{1}{|B_{2R} |}\int _{B_{2R}}(w^1)^p dx \Big) ^{1/p} \] where $c>0$ depends on $n$, $p$. Since $w^1\in C^{2+\alpha }$ near the boundary and $w^1= e^1\ge \cos r >0$ on $\partial M $, there exists $\delta >0$ such that if we define $M _{\delta }=\{ x\in M ; \operatorname{dist} (x, \partial M ) \le \delta \}$, then $w^1\ge c_0:=\cos r /2$ in $M _{\delta }$. Since $\dim _H\operatorname{sing}(w^1)\le n-3 $ (in the case where $n=3$, $\operatorname{sing}(w^1)$ is discrete), for any $x_0 \in M \setminus \operatorname{sing}(w^1)$, we can choose $x_1\in M _{\delta }$ and a continuous curve $l$ in $M $ joining $x_0$ and $x_1$ such that $l \cap \operatorname{sing}(w^1)=\emptyset $. For every $x\in l$, there exists $R>0$ such that $B_{2R}(x)$ is contained in a local coordinate neighborhood and \begin{equation} \operatorname{ess\,inf} _{B_R(x)}w^1 \ge c \Big( \frac{1}{|B_{2R}(x) |} \int _{B_{2R}(x)} (w^1)^p dx \Big)^{1/p}. \label{e3.4} \end{equation} Since $l$ is compact, there exist finitely many $R_j>0$ and $ x_{(j)}\in l$ $(j=1,2,\ldots ,N)$ such that $\cup _{j=1}^N B_{R_j}(x_{(j)})\supset l$ and $x_{(1)}=x_0, x_{(N)}=x_1$. Since $\operatorname{ess\,inf} _{B_R(x_{(N)})}w^1>0$, it follows from \eqref{e3.4} that $\operatorname{ess\,inf} _{B_R(x_{(N-1)})}w^1>0$. Repeating this procedure, we have $\operatorname{ess\,inf} _{B_R(x_0)}w^1>0$. In particular, $w^1(x_0)>0$. Thus we see that $w^1>0$ in $M \setminus \operatorname{sing}(w^1)$. Hence we see that $u^1>0$ in $M \setminus \operatorname{sing}(u^1)$ or $u^1<0$ in $M \setminus \operatorname{sing}(u^1)$. Since $u^1=e_1>0 $ on $\partial M $, we have $u^1>0$ in $M \setminus \operatorname{sing}(u^1)$. Since $u^1$ is continuous near $\partial M $, there exist $\delta >0$ and $c_0>0$ such that $u^1\ge c_0$ on $M _{\delta }$. Define $M ^{\delta }=\{x \in M ; \operatorname{dist} (x,\partial M )\ge \delta \}$. Choose $R>0$ so that $2R< \delta $ and fix $1\le p0$ such that for any $B_{2R}(y) $ contained in a local coordinate neighborhood, \[ \operatorname{ess\,inf} _{B_R(y)}u^1 \ge c' \Big( \frac{1}{|B_{2R}(y)|}\int _{B_{2R}(y)}(u^1)^p dv_g \Big)^{1/p}. \] Since $M ^{\delta }$ is compact, there exists finitely many points $y_i$ and positive numbers $R_i \, (i=1,2,\ldots ,L)$ such that $\cup _{i=1}^L B_{R_i}(y_i)\supset M ^{\delta }$. If we define \[ c_i = c_i'\Big(\frac{1}{|B_{2R_i}(y_i)|} \int _{B_{2R_i}(y_i)}(u^1)^p dx \Big)^{1/p} \quad (i=1,2,\ldots ,L), \] and $c= \min \{ c_0,c_1,\ldots ,c_L\}$, we have $u^1\ge c$ a.e. on $M$. Therefore we can find $r'$ with $r