A common problem in physics are the time harmonic Maxwell equations. Faraday's law and Ampere's law can be combined to form $$ \nabla \times (\nabla \times \mathbf{E}) -\frac{\omega^2}{c^2} \mathbf{E} = i\omega \mu_0 \mathbf{J} $$ With E the electric field, J the current density, and the rest constants. Let's say that J is given in the domain. Let's assume the volume is 3D. A common boundary condition is a PEC (perfect electric conductor), which means that on boundary $\partial\Omega$: $$\mathbf{n}\times \mathbf{E} = \mathbf{0}$$ With $\mathbf{n}$ the unit normal vector on the wall. In other words, the components of E tangent to the wall are short circuited, so 0. This can be solved using finite elements, for this reason I will write it in the weak form, so taking the dot product with an arbitrary vector $\mathbf{F}^*$, and integrate over the volume $\Omega$. The asterisk indicates the complex conjugate. $$ \int_\Omega \left\{(\nabla \times \mathbf{F}^*)\cdot (\nabla \times \mathbf{E})-\frac{\omega^2}{c^2} \mathbf{F}^*\cdot \mathbf{E}\right\}dV = \int_{\partial \Omega} \mathbf{n}\cdot\left\{\mathbf{F}^* \times (\nabla \times \mathbf{E}) \right\}dS + \int_\Omega i\omega \mu_0 \mathbf{F}^*\cdot \mathbf{J} dV $$ The surface integral can be eliminated using the boundary conditions: $$ \mathbf{n}\cdot\left\{\mathbf{F}^* \times (\nabla \times \mathbf{E}) \right\} = (\nabla \times \mathbf{E})\cdot (\mathbf{n}\times \mathbf{F}^*) = 0 $$ Because we choose $\mathbf{F}$ to have the same boundary conditions as $\mathbf{E}$.
We finally get to my question, what I don't understand is how this problem can be well posed with just a boundary condition on the tangential components of $\mathbf{E}$. No restriction is imposed on the normal component. So there are 3 unknown quantities (three components of $\mathbf{E}$), yet only 2 boundary conditions.
Using the interface conditions for EM fields a 3rd condition can be derived for the normal component, based on the surface charge density on the boundary wall. However, this charge is usually not know in practice. Moreover, many of the common software packages (e.g. comsol) use a PEC without mentioning surface charge. This leads me to believe that you do not need 3 boundary conditions, is this true?