2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60662New extrinsic lower bounds are given for the classical Dirac operator on the boundary of a compact domain of a spin manifold. The main tool is to solve some boundary problems for the Dirac operator of the domain under boundary conditions of Atiyah-Patodi-Singer type. Spinorial techniques are used to give simple proofs of classical results for compact embedded hypersurfaces.16 pages, LaTex, to appear in Math. Res. LettersDifferential GeometryMathematical Physics53C27;53C40;53C80;58G25text