\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil An $\epsilon$-regularity result \hfil EJDE--2003/01} {EJDE--2003/01\hfil Roger Moser \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2003}(2003), No. 01, pp. 1--7. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % An $\epsilon$-regularity result for generalized harmonic maps into spheres % \thanks{ {\em Mathematics Subject Classifications:} 58E20. \hfil\break\indent {\em Key words:} Generalized harmonic maps, regularity. \hfil\break\indent \copyright 2003 Southwest Texas State University. \hfil\break\indent Submitted December 13, 2002. Published January 2, 2003.} } \date{} % \author{Roger Moser} \maketitle \begin{abstract} For $m,n \ge 2$ and $1 < p < 2$, we prove that a map $u \in W_\mathrm{loc}^{1,p}(\Omega,\mathbb{S}^{n - 1})$ from an open domain $\Omega \subset \mathbb{R}^m$ into the unit $(n - 1)$-sphere, which solves a generalized version of the harmonic map equation, is smooth, provided that $2 - p$ and $[u]_{\mathrm{BMO}(\Omega)}$ are both sufficiently small. This extends a result of Almeida \cite{almeida95}. The proof is based on an inverse H\"older inequality technique. \end{abstract} \newcommand{\scp}[2]{\left\langle #1, #2 \right\rangle} \newcommand{\set}[2]{\{ #1 \, \colon \ #2 \}} \newcommand{\intbar}{{- \hspace{- 1.05 em}} \int} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} For integers $m,n \ge 2$, let $\Omega \subset \mathbb{R}^m$ be an open domain, and let $\mathbb{S}^{n - 1} \subset \mathbb{R}^n$ denote the $(n - 1)$-dimensional unit sphere. Define the space \[ H^1(\Omega,\mathbb{S}^{n - 1}) = \set{v \in H^1(\Omega,\mathbb{R}^n)}{|v| = 1 \mbox{ almost everywhere}}, \] and consider the functional \[ E(u) = \frac{1}{2} \int_\Omega |\nabla u|^2 \, dx, \quad u \in H^1(\Omega, \mathbb{S}^{n - 1}). \] A map $u \in H^1(\Omega,\mathbb{S}^{n - 1})$ is called a weakly harmonic map, if it is a critical point of $E$, i.~e.\ \[ \frac{d}{dt}\Big|_{t = 0} E\big(\frac{u + t\phi}{|u + t \phi|}\big) = 0 \] for all $\phi \in C_0^\infty(\Omega,\mathbb{R}^n)$. The Euler-Lagrange equation for this variational problem is \begin{equation} \label{EL} \Delta u + |\nabla u|^2 u = 0 \quad \mbox{in } \Omega \end{equation} (in the distributions sense). Denote by $\wedge$ the exterior product $\wedge : \mathbb{R}^n \times \mathbb{R}^n \to \Lambda_2 \mathbb{R}^n$, then (\ref{EL}) is equivalent to \begin{equation} \label{EL2} \mathop{\rm div} (u \wedge \nabla u) = 0 \quad \mbox{in } \Omega. \end{equation} This form of the equation provides a natural extension of the notion of weakly harmonic maps into spheres. Whereas we need a map in $H_\mathrm{loc}^1(\Omega, \mathbb{S}^{n - 1})$ to make any sense of (\ref{EL}), the equation (\ref{EL2}) only requires \[ u \in W_\mathrm{loc}^{1,1}(\Omega,\mathbb{S}^{n - 1}) = \set{v \in W_\mathrm{loc}^{1,1}(\Omega,\mathbb{R}^n)}{|v| = 1 \mbox{ almost everywhere}}. \] A map in this space satisfying (\ref{EL2}) is called a generalized harmonic map. For $m = 2$, it was proven by H\'elein \cite{helein90,helein91}, that any weakly harmonic map is smooth (also for more general target manifolds than spheres). For higher dimensions, this is no longer true. Indeed Rivi\`ere \cite{riviere95} constructed a weakly harmonic map in three dimensions which is discontinuous everywhere. But there exists an $\epsilon$-regularity result, due to Evans \cite{evans91} (and to Bethuel \cite{bethuel93} for more general targets), which can be stated as follows. \begin{theorem} \label{thme} There exists a number $\epsilon > 0$, depending only on $m$ and $n$, such that any weakly harmonic map $u \in H^1(\Omega,\mathbb{S}^{n - 1})$ with the property $[u]_{\mathrm{BMO}(\Omega)} \le \epsilon$ is smooth in $\Omega$. \end{theorem} Here we use the notation \begin{equation} \label{er} [u]_{\mathrm{BMO}(\Omega)} = \sup_{B_r(x_0) \subset \Omega} \, \intbar_{B_r(x_0)} |u - \bar{u}_{B_r(x_0)}| \, dx, \end{equation} where $B_r(x_0)$ denotes the ball in $\mathbb{R}^m$ with centre $x_0$ and radius $r$, and \[ \bar{u}_{B_r(x_0)} = \intbar_{B_r(x_0)} u \, dx = \frac{1}{|B_r(x_0)|} \int_{B_r(x_0)} u \, dx. \] Together with the well-known monotonicity formula for so-called stationary weakly harmonic maps, e.~g.\ weakly harmonic maps which satisfy $\frac{d}{dt}|_{t = 0} E(u(x + t \psi(x))) = 0$ for all $\psi \in C_0^\infty(\Omega,\mathbb{R}^m)$ (see Price \cite{price83}), one concludes that weakly harmonic maps with this property are smooth away from a closed singular set of vanishing $(m - 2)$-dimensional Hausdorff measure. Generalized harmonic maps on the other hand may have singularities even in two dimensions. A typical example is the map $u(x) = x/|x|$ in $\mathbb{R}^2$. For $m = 2$ and for any $p \in [1,2)$, Almeida \cite{almeida95} even constructed generalized harmonic maps in $W^{1,p}(\Omega,\mathbb{S}^1)$ which are nowhere continuous. Nevertheless, there is an $\epsilon$-regularity result for generalized harmonic maps in two dimensions, due to Almeida \cite{almeida95}. (Another proof was given by Ge \cite{ge99}.) \begin{theorem} For $m = 2$, there exists $\epsilon > 0$, depending only on $n$, such that any weakly harmonic map $u \in W_\mathrm{loc}^{1,1}(\Omega,\mathbb{S}^{n - 1})$ with the property $\|\nabla u\|_{L^{2,\infty}(\Omega)} \le \epsilon$ is smooth in $\Omega$. \end{theorem} Here $\|\cdot\|_{L^{2,\infty}(\Omega)}$ is the norm of the Lorentz space $L^{2,\infty}(\Omega,\mathbb{R}^{m \times n})$. (For a definition and properties of Lorentz spaces, see e.~g.\ \cite{steinweiss71}, Chapter V.) \section{Results} The aim of this note is to extend and improve this result. We replace the smallness in the $L^{2,\infty}$-norm by a weaker condition (reminding of Theorem \ref{thme}), and we prove the result for all dimensions. More precisely, we have the following theorem. \begin{theorem} \label{thm1} There exist $p < 2$ and $\epsilon > 0$, depending only on $m$ and $n$, such that any generalized harmonic map $u \in W_\mathrm{loc}^{1,p}(\Omega,\mathbb{S}^{n - 1})$ with the property $[u]_{\mathrm{BMO}(\Omega)} \le \epsilon$ is in $C^\infty(\Omega,\mathbb{S}^{n - 1})$. \end{theorem} To prove this theorem, it suffices to show that under these conditions, the generalized harmonic map $u$ is in $H_\mathrm{loc}^1(\Omega, \mathbb{S}^{n - 1})$. Higher regularity is then implied by Theorem \ref{thme} (provided that $\epsilon$ is chosen accordingly). For this first step on the other hand, we can also admit a non-vanishing right hand side in (\ref{EL2}). \begin{theorem} \label{thm2} For any $q > 2$, there exist $p < 2$ and $\epsilon > 0$, depending only on $m$, $n$, and $q$, with the following property. Suppose that $u \in W_\mathrm{loc}^{1,p}(\Omega,\mathbb{S}^{n - 1})$ is a distributional solution of \begin{equation} \label{nh} \mathop{\rm div}(u \wedge \nabla u) = F + \mathop{\rm div} G, \end{equation} where $F \in L_\mathrm{loc}^{mq/(m + q)}(\Omega,\Lambda_2 \mathbb{R}^n)$ and $G \in L_\mathrm{loc}^q(\Omega,\mathbb{R}^m \otimes \Lambda_2 \mathbb{R}^n)$. If $[u]_{\mathrm{BMO}(\Omega)} \le \epsilon$, then $u \in W_\mathrm{loc}^{1,p/(p - 1)}(\Omega,\mathbb{S}^{n - 1})$. \end{theorem} As mentioned above, Theorem \ref{thm1} is an immediate consequence of Theorem \ref{thme} and Theorem \ref{thm2}. The proof of the latter is inspired by the inverse H\"older inequality technique used by Iwaniec--Sbordone \cite{iwaniecsbordone94} to prove regularity for solutions of equations of the form \[ \mathop{\rm div} A(x,\nabla u) = F + \mathop{\rm div} G, \] where $A(x,\xi) = \frac{\partial\mathcal{F}}{\partial\xi}(x,\xi)$ for a quasi-convex function $\mathcal{F}$ (satisfying certain conditions). We combine these methods with arguments from the regularity theory for weakly harmonic maps. We will use the following well-known results. The first one is due to Gia\-quin\-ta--Modica \cite{giaquintamodica79}. \begin{proposition} \label{prop1} For $1 < a < b$, and for some ball $B_R(x_0) \subset \mathbb{R}^m$, suppose that $g \in L^a(B_R(x_0))$ and $f \in L^b(B_R(x_0))$ are non-negative functions which satisfy \[ \intbar_{B_{r/2}(x_1)} g^a \, dx \le A\Big[\Big(\intbar_{B_r(x_1)} g \, dx\Big)^a + \intbar_{B_r(x_1)} f^a \, dx\Big] + \theta \intbar_{B_r(x_1)} g^a \, dx \] for every ball $B_r(x_1) \subset B_R(x_0)$ and for certain constants $A,\theta > 0$. There exists a constant $\theta_0 = \theta_0(m,a,b) > 0$, such that whenever $\theta < \theta_0$, then $g \in L^c(B_{R/2}(x_0))$ with \[ \Big(\intbar_{B_{R/2}(x_0)} g^c \, dx\Big)^{1/c} \le B \Big[\Big(\intbar_{B_R(x_0)} g^a \, dx\Big)^{1/a} + \Big(\intbar_{B_R(x_0)} f^c \, dx\Big)^{1/c}\Big] \] for certain numbers $c > a$ and $B > 0$, both depending only on $m$, $A$, $\theta$, $a$, and $b$. \end{proposition} The following is a combination of the compensated compactness results of Coifman--Lions--Meyer--Semmes \cite{coifmanlionsmeyersemmes93}, and the duality of the space $\mathrm{BMO}(\mathbb{R}^m) = \set{f \in L_\mathrm{loc}^1(\mathbb{R}^m)}{[f]_{\mathrm{BMO}(\mathbb{R}^m)} < \infty}$ with the Hardy space $\mathcal{H}^1(\mathbb{R}^m)$. The latter is due to Fefferman--Stein \cite{feffermanstein72}. \begin{proposition} \label{prop2} For $1 < p < \infty$, suppose that a function $f \in W_\mathrm{loc}^{1,p}(\mathbb{R}^m)$ with $\|\nabla f\|_{L^p(\mathbb{R}^m)} < \infty$, a vector field $g \in L^{p/(p - 1)}(\mathbb{R}^m, \mathbb{R}^m)$ with $\mathop{\rm div} g = 0$ in the distribution sense, and a function $h \in \mathrm{BMO}(\mathbb{R}^m)$ are given. Then \[ \Big|\int_{\mathbb{R}^m} \nabla f \cdot g \, h \, dx\Big| \le C \|\nabla f\|_{L^p(\mathbb{R}^m)} \|g\|_{L^{p/(p - 1)}(\mathbb{R}^m)} [h]_{\mathrm{BMO}(\mathbb{R}^m)} \] for a constant $C$ which depends only on $m$ and $p$. \end{proposition} Having the ingredients ready, we can now prove Theorem \ref{thm2}. \noindent{\it Proof of Theorem \ref{thm2}.} Suppose $q > 2$, $F \in L_\mathrm{loc}^{mq/(m + q)}(\Omega,\Lambda_2 \mathbb{R}^n)$, and $G \in L_\mathrm{loc}^q(\Omega,\mathbb{R}^m \otimes \Lambda_2 \mathbb{R}^n)$. Let for the moment $p$ be any number in $(1,2)$, and suppose that $u \in W_\mathrm{loc}^{1,p}(\Omega,\mathbb{S}^{n - 1})$ is a solution of (\ref{nh}). Let $\psi \in W_\mathrm{loc}^{2,mq/(m + q)}(\Omega,\Lambda_2\mathbb{R}^n)$ be a solution of \[ \Delta \psi = F \quad \mbox{in } \Omega. \] Then $\nabla \psi \in W_\mathrm{loc}^{1,q}(\Omega,\mathbb{R}^m \otimes \Lambda_2\mathbb{R}^n)$, and $u$ satisfies \[ \mathop{\rm div}(u \wedge \nabla u) = \mathop{\rm div} (G + \nabla \psi). \] Hence we may assume without loss of generality that $F = 0$. Choose a ball $B_r(x_0) \subset \Omega$ and a cut-off function $\zeta \in C_0^\infty(B_r(x_0))$ with $\zeta \equiv 1$ in $B_{r/2}(x_0)$, such that $|\nabla \zeta| \le 4r^{-1}$. Consider the Hodge decomposition \[ |\nabla (\zeta(u - \bar{u}_{B_r(x_0)}))|^{p - 2} \, u \wedge \nabla (\zeta(u - \bar{u}_{B_r(x_0)})) = \nabla \phi + \Phi, \] where $\phi \in W_\mathrm{loc}^{1,p/(p - 1)}(\mathbb{R}^m,\Lambda_2 \mathbb{R}^n)$ and $\Phi \in L^{p/(p - 1)}(\mathbb{R}^m,\mathbb{R}^m \otimes \Lambda_2 \mathbb{R}^n)$ have the properties $\mathop{\rm div} \Phi = 0$ and \[ \|\nabla \phi\|_{L^s(\mathbb{R}^m)} + \|\Phi\|_{L^s(\mathbb{R}^m)} \le C_1 \|\nabla (\zeta(u - \bar{u}_{B_r(x_0)}))\|_{L^{(p - 1)s}(B_r(x_0))}^{p - 1} \] for any $s \in (\frac{1}{p - 1},\frac{p}{p - 1}]$ and for a constant $C_1 = C_1(m,n,s)$. The existence of such a decomposition is due to Iwaniec--Martin \cite{iwaniecmartin93}. In particular, we have \begin{equation} \label{p2} \intbar_{B_r(x_0)} |\nabla \phi|^s \, dx \le C_2 \left(\intbar_{B_r(x_0)} |\nabla u|^s \, dx\right)^{p - 1} \end{equation} for a constant $C_2 = C_2(m,n,s)$, owing to the Poincar\'e and the H\"older inequality. Observe that \begin{eqnarray*} 2^{-m} \intbar_{B_{r/2}(x_0)} |\nabla u|^p \, dx & \le & \intbar_{B_r(x_0)} \scp{u \wedge \nabla (\zeta(u - \bar{u}_{B_r(x_0)}))}{\nabla \phi + \Phi} \, dx \\ & = & \intbar_{B_r(x_0)} \scp{u \wedge \nabla (\zeta(u - \bar{u}_{B_r(x_0)}))}{\Phi} \, dx \\ && {} + \intbar_{B_r(x_0)} \scp{\nabla \zeta}{(u \wedge (u - \bar{u}_{B_r(x_0)})) \cdot \nabla \phi} \, dx \\ && {} - \intbar_{B_r(x_0)} \scp{\nabla \zeta}{(u \wedge \nabla u) \cdot (\phi - \bar{\phi}_{B_r(x_0)})} \, dx \\ && {} + \intbar_{B_r(x_0)} \scp{G}{\nabla (\zeta (\phi - \bar{\phi}_{B_r(x_0)}))} \, dx, \end{eqnarray*} where we denote the standard scalar product in $\mathbb{R}^m$ and in $\mathbb{R}^m \otimes \Lambda_2 \mathbb{R}^n$ by $\scp{\cdot}{\cdot}$, whereas we use a dot in $\mathbb{R}^n$ to avoid confusion. We have the estimates \begin{eqnarray*} \lefteqn{\intbar_{B_r(x_0)} \scp{\nabla \zeta}{(u \wedge (u - \bar{u}_{B_r(x_0)})) \cdot \nabla \phi} \, dx} \\ & \le & \frac{4}{r}\Big(\intbar_{B_r(x_0)} |\nabla \phi|^\frac{2m}{m + 1} \, dx\Big)^\frac{m + 1}{2m} \Big(\intbar_{B_r(x_0)} |u - \bar{u}_{B_r(x_0)}|^\frac{2m}{m - 1} \, dx\Big)^\frac{m - 1}{2m} \\ & \le & C_3 \Big(\intbar_{B_r(x_0)} |\nabla u|^\frac{2m}{m + 1} \, dx \Big)^\frac{p(m + 1)}{2m}, \end{eqnarray*} by (\ref{p2}) and the Sobolev inequality, and similarly \[ - \intbar_{B_r(x_0)} \scp{\nabla \zeta}{(u \wedge \nabla u) \cdot (\phi - \bar{\phi}_{B_r(x_0)})} \, dx \le C_4 \left(\intbar_{B_r(x_0)} |\nabla u|^\frac{2m}{m + 1} \, dx \right)^\frac{p(m + 1)}{2m}, \] for certain constants $C_3,C_4$ which depend only on $m$ and $n$. Note that $[\zeta(u - \bar{u}_{B_r(x_0)})]_{\mathrm{BMO}(\mathbb{R}^m)} \le C_5 [u]_{\mathrm{BMO}(B_r(x_0))}$ for a constant $C_5 = C_5(m,n)$. (This is proven in \cite{evans91}.) Extending $\nabla u$ to $\mathbb{R}^m$ and applying Proposition \ref{prop2}, we thus find \begin{eqnarray*} \lefteqn{ \intbar_{B_r(x_0)} \scp{u \wedge \nabla (\zeta(u - \bar{u}_{B_r(x_0)}))}{\Phi} \, dx }\\ & = & - \intbar_{B_r(x_0)} \zeta \scp{\nabla u \wedge (u - \bar{u}_{B_r(x_0)})}{\Phi} \, dx \\ & \le & C_6 [u]_{\mathrm{BMO}(\Omega)} \intbar_{B_r(x_0)} |\nabla u|^p \, dx \end{eqnarray*} for a constant $C_6 = C_6(m,n,p)$. Finally, choose a number $\sigma \in (2,q)$. We have \begin{eqnarray*} \lefteqn{\intbar_{B_r(x_0)} \scp{G}{\nabla (\zeta(\phi - \bar{\phi}_{B_r(x_0)}))} \, dx} \\ & \le & C_7 \Big(\intbar_{B_r(x_0)} |G|^\sigma \, dx\Big)^{1/\sigma} \Big(\intbar_{B_r(x_0)} |\nabla \phi|^{\sigma/(\sigma - 1)}\, dx \Big)^\frac{\sigma - 1}{\sigma} \\ & \le & C_8 \Big(\intbar_{B_r(x_0)} |G|^\sigma \, dx\Big)^{1/\sigma} \Big(\intbar_{B_r(x_0)} |\nabla u|^{\sigma/(\sigma - 1)}\, dx \Big)^\frac{(p - 1)(\sigma - 1)}{\sigma} \\ & \le & C_8 \Big[\intbar_{B_r(x_0)} |G|^\sigma \, dx + \Big(\intbar_{B_r(x_0)} |\nabla u|^{\sigma/(\sigma - 1)} \, dx\Big)^\frac{p(\sigma - 1)}{\sigma} + 1\Big] \end{eqnarray*} (for constants $C_7,C_8$ which depend on $m$, $n$, and $\sigma$) by the H\"older inequality, the Poincar\'e inequality, the estimate (\ref{p2}), and Young's inequality. Now choose $a \in (1,\min\{\frac{m + 1}{m},\frac{2(\sigma - 1)}{\sigma}\})$, and set $b = \frac{qa}{\sigma}$. Let $\theta_0$ be the constant from Proposition \ref{prop1} (belonging to $a$ and $b$), and choose a number $\theta \in (0,\theta_0)$. Then the conditions of Proposition \ref{prop1} are satisfied for any ball $B_R(x_0) \subset\subset \Omega$, for the functions \[ g = |\nabla u|^{p/a}, \quad f = |G|^{\sigma/a} + 1, \] and for a constant $A$ which depends only on $m$, $n$, and $\sigma$, provided that $p \ge a \max\{\frac{2m}{m + 1}, \frac{\sigma}{\sigma - 1}\}$ (which is strictly less than $2$) and $[u]_{\mathrm{BMO}(\Omega)} \le C_6^{-1} \theta$. Hence under these conditions, there exists a number $c > a$, not depending on $p$, such that $|\nabla u| \in L_\mathrm{loc}^{pc/a}(\Omega)$. If $2 - p$ is sufficiently small, then $\frac{pc}{a} \ge \frac{p}{p - 1}$, and therefore $u \in W_\mathrm{loc}^{1,p/(p - 1)}(\Omega,\mathbb{S}^{n - 1})$. 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