  We investigate the self-adjointness of the two-dimensional Dirac operator
$D$, with quantum-dot and Lorentz-scalar $\delta$-shell boundary conditions, on
piecewise $C^2$ domains with finitely many corners. For both models, we prove
the existence of a unique self-adjoint realization whose domain is included in
the Sobolev space $H^{1/2}$, the formal form domain of the free Dirac operator.
The main part of our paper consists of a description of the domain of $D^*$ in
terms of the domain of $D$ and the set of harmonic functions that verify some
mixed boundary conditions. Then, we give a detailed study of the problem on an
infinite sector, where explicit computations can be made: we find the
self-adjoint extensions for this case. The result is then translated to general
domains by a coordinate transformation.
