Self-adjointness of two-dimensional Dirac operators on corner domains

We investigate the self-adjointness of 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 existence a unique self-adjoint realization whose domain is included in Sobolev space $H^{1/2}$, formal form free operator. The main part our paper consists description adjoint $D^\*$ terms $D$ set harmonic functions that verify some $mixed$ conditions. Then, give detailed study problem an infinite sector, where explicit computations can be made: find extensions for this case. result then translated to general by coordinate transformation.