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.
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Acknowledgments
This work was started whilst both authors were supported by The Leverhulme Trust at the University of Warwick. The second author has received funding from the European Research Council under the European Union—Seventh Framework Programme (FP7/2007-2013)/ERC Arant Agreement No. 267087. The first author has also received funding from the European Research Council STG agreement number P34897. The first author would like to thank Jürgen Jost and the Max Plank Institute for Mathematics in the Sciences for their hospitality during the completion of this work.
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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
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
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
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
Theorem 4.8
Let \(k\in \mathbb {N}_0\) and \(1<p<\infty \). Suppose that \(u\in W^{1,p}\) weakly solves
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
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
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
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Sharp, B., Zhu, M. Regularity at the free boundary for Dirac-harmonic maps from surfaces. Calc. Var. 55, 27 (2016). https://doi.org/10.1007/s00526-016-0960-4
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DOI: https://doi.org/10.1007/s00526-016-0960-4