The Hijazi inequality on manifolds with boundary
  In this paper, we prove the Hijazi inequality on compact Riemannian spin manifolds under two boundary conditions: the condition associated with a chirality operator and the Riemannian version of the $\MIT$ bag condition. We then show that the limiting-case is characterized as being a half-sphere for the first condition whereas the equality cannot be achieved for the second. 
Simon Raulot (IECN)