The maximum principle and the Dirichlet problem for Dirac-harmonic maps (Q1949414)

Dirac-harmonic maps are the critical points of a functional that couples the nonlinear sigma model with a spinor field. The functional is motivated by the supersymmetric version of the sigma model. Dirac-harmonic maps have been introduced in [\textit{Q. Chen} et al., Math. Z. 254, No. 2, 409--432 (2006; Zbl 1103.53033)]. While issues like regularity, blow-up analysis and even some partial classification have been studied before, the general existence problem has no satisfying solution yet. A step towards that is made in the current paper, which is concerned with Dirac-harmonic maps from a Riemannian spin manifold with boundary into a regular ball in any Riemannian manifold. The restriction to regular balls allows to generalize a maximum principle for harmonic maps by \textit{W. Jäger} and \textit{H. Kaul} [Manuscr. Math. 28, 269--291 (1979; Zbl 0413.31006)] to the Dirac-harmonic case. The maximum principle helps to derive a priori estimates, which in turn give rise to an existence and uniqueness theorem for Dirac-harmonic maps with small boundary data.