We study spectral properties of Dirac operators on bounded domains $Ω\subset \mathbb{R}^3$ with boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter $τ\in\mathbb{R}$; the case $τ= 0$ corresponds to the MIT bag model. We show that the eigenvalues are parametrized as increasing functions of $τ$, and we exploit this monotonicity to study the limits as $τ\to \pm \infty$. We prove that if $Ω$ is not a ball then the first positive eigenvalue is greater than the one of a ball with the same volume for all $τ$ large enough. Moreover, we show that the first positive eigenvalue converges to the mass of the particle as $τ\downarrow -\infty$, and we also analyze its first order asymptotics.

49 pages, 5 figures. v2: version after referee report (Conjecture 1.8 and Remark 1.9 added) v3: Final version