"This talk aims to present estimates on the lowest eigenvalue in the gap of a Dirac operator in terms of a Lebesgue norm of the potential. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. Most of our result are established in the Birman-Schwinger reformulation of the problem. This is a collaboration work with Jean Dolbeault and David Gontier (University Paris-Dauphine), and Hanne Van Den Bosch (University of Chile)."