Regularity at the free boundary for Dirac-harmonic maps from surfaces

Abstract

We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our methods also lead to the full interior $\epsilon$-regularity and smooth estimates for weakly Dirac-harmonic maps in all dimensions.

Appendices

Appendix 1: Morrey spaces

We introduce the Morrey spaces $M^{p,\beta }(E)$ for $1\le p<\infty$ and $0\le \beta \le m$ ($E \subset \mathbb {R}^m$). We say that $g \in M^{p,\beta }(E)$ if $M_{\beta }[g^p](x):= \sup _{r>0} r^{-\beta } \int _{B_r(x)\cap E} |g|^p \in L^{\infty }$ with norm (which makes $M^{p,\beta }$ a Banach space) $\Vert g\Vert _{M^{p,\beta }(E)}^p = \Vert M_{\beta }[g^p]\Vert _{L^{\infty }(E)}.$

We note the following extension theorem for Sobolev-Morrey spaces:

Theorem 4.6

The unit ball $B_1\subset \mathbb {R}^m$ is an extension domain for $M_1^{p,\beta }$. Given $v\in M_1^{p,\beta }(B_1)$ there exists ${\tilde{v}}\in M_1^{p,\beta }(\mathbb {R}^m)$ such that

$$\begin{aligned} \Vert {\tilde{v}}\Vert _{M^{p,\beta }} + \Vert {\nabla }{\tilde{v}}\Vert _{M^{p,\beta }} \le C\left( \Vert v\Vert _{M^{p,\beta }(B_1)} + \Vert {\nabla }v\Vert _{M^{p,\beta }(B_1)}\right) \end{aligned}$$

and $v=\tilde{v}$ almost everywhere.

We could not find a proof of this theorem, however it follows easily by standard techniques for extension theorems. In general it should remain true for any domain U with $\partial U$ as above but $\phi$ must be Lipschitz.

We also note that if $\int _{B_1} v = 0$ then

$$\begin{aligned} \Vert {\nabla }{\tilde{v}}\Vert _{M^{p,\beta }} \le C\left( [v]_{\mathcal {C}^{p,\beta }(B_1)} + \Vert {\nabla }v\Vert _{M^{p,\beta }(B_1)}\right) \le C\Vert {\nabla }v\Vert _{M^{p,\beta }(B_1)} \end{aligned}$$

where $\mathcal {C}^{p,\beta }$ is the Campanato space—see [10].

Appendix 2: Classical boundary value estimates

Here we recall results about the classical boundary value problems for the Laplacian and the Cauchy-Riemann operator, we refer the reader to [2, 3, 35] for background material. For all of the theorems below $U\subset \mathbb {R}^m$ is any open domain and $T\subset \partial U$ is a smooth boundary portion.

Theorem 4.7

Let $k\in \mathbb {N}_0$ and $1<p<\infty$. Suppose that $u\in W^{1,p}$ weakly solves

$$\begin{aligned} -{\Delta }u= & {} f\in W^{k,p}(U), \end{aligned}$$

(4.8)

$$\begin{aligned} u= & {} 0 \quad \text {on }T \end{aligned}$$

(4.9)

then for any $V\subset \subset U\cup T$, $u\in W^{k+2, p}(V)$ and there exists some $C=C(p,k,V,T)>0$ such that

$$\begin{aligned} \Vert u\Vert _{W^{k+2,p}(V)} \le C\left( \Vert f\Vert _{W^{k,p}(U)} +\Vert u\Vert _{L^p(U)}\right) . \end{aligned}$$

Theorem 4.8

Let $k\in \mathbb {N}_0$ and $1<p<\infty$. Suppose that $u\in W^{1,p}$ weakly solves

$$\begin{aligned} -{\Delta }u= & {} f\in W^{k,p}(U), \end{aligned}$$

(4.10)

$$\begin{aligned} {\frac{\partial u}{\partial \overrightarrow{n}}}= & {} g \in W_{\partial }^{k+1,p}(T) \end{aligned}$$

(4.11)

then for any $V \subset \subset U\cup T$, $u\in W^{k+2, p}(V)$ and there exists some $C=C(p,k,V,T)>0$ such that

$$\begin{aligned} \Vert u\Vert _{W^{k+2,p}(V)} \le C\left( \Vert f\Vert _{W^{k,p}(U)}+\Vert g\Vert _{W_{\partial }^{k+1,p}(T)} + \Vert u\Vert _{L^p(U)}\right) . \end{aligned}$$

We also recall the analogue for the Cauchy-Riemann operator in $\mathbb {C}$.

Theorem 4.9

Let $U\subset \mathbb {C}$ be any domain and $T\subset \partial U$ a smooth boundary portion, $k\in \mathbb {N}_0$ and $1<p<\infty$. Suppose that $h\in W^{1,p}$ solves

$$\begin{aligned} \overline{\partial }h = f\in W^{k,p}(U), \end{aligned}$$

(4.12)

$$\begin{aligned} Re(h) =0 \quad \text {or}\quad Im(h) = 0 \quad \text {on }T \end{aligned}$$

(4.13)

then for any $V\subset \subset U\cup T$, $h\in W^{k+1, p}(V)$ and there exists some $C=C(p,k,V,T)>0$ such that

$$\begin{aligned} \Vert h\Vert _{W^{k+1,p}(V)} \le C\left( \Vert f\Vert _{W^{k,p}(U)} +\Vert h\Vert _{L^{p}(U)}\right) . \end{aligned}$$