Geometric analysis of a mixed elliptic-parabolic conformally invariant boundary value problem

We treat the blow-up behavior of α-Dirac-harmonic maps.Such maps arise from a version of the nonlinear supersymmetric σ-model of QFT, and the difficulties arise from the conformal invariance of the problem and the coupling of a second order elliptic (or parabolic) system for a map with a first order elliptic system for a spinor along this map.The solutions are called Dirac-harmonic maps.Building upon the pioneering work of Sacks-Uhlenbeck, we develop a general spectrum of methods (Pohozaev identity, three circle method, blow-up analysis, energy identities, energy decay estimates etc.) for conformally invariant variational problems at this particularly challenging example.We study the refined blow-up behaviour and asymptotic analysis for a sequence of α-Dirac harmonic maps from a compact Riemann surface with smooth boundary into a compact Riemannian manifold with uniformly bounded energy.We prove generalized energy identities and novel energy decay estimates.This enables us to present a very detailed blow-up analysis.