  We develop estimates for the solutions and derive existence and uniqueness
results of various local boundary value problems for Dirac equations that
improve all relevant results known in the literature. With these estimates at
hand, we derive a general existence, uniqueness and regularity theorem for
solutions of Dirac equations with such boundary conditions. We also apply these
estimates to a new nonlinear elliptic-parabolic problem, the Dirac-harmonic
heat flow on Riemannian spin manifolds. This problem is motivated by the
supersymmetric nonlinear $\sigma$-model and combines a harmonic heat flow type
equation with a Dirac equation that depends nonlinearly on the flow.
