If $B$ is magnetic field and $E$ electric Field, then
$$B=\nabla\times A,$$
$$E= -\nabla V+\frac{\partial A}{\partial t}.$$
There is Gauge invariance for the transformation
$$A'\rightarrow A+{\nabla L}$$
$$V'\rightarrow V-\frac{dL}{dt}.$$
Now, we can write:
Coulomb Gauge (CG): the choice of a $L$ that implies $\nabla\cdot A=0$.
Lorenz Gauge (LG): the choice of a $L$ that implies $\nabla \cdot A - \frac{1}{c^2} \frac{\partial V}{\partial t}=0$.
Now, I'm trying to mathematically prove that it's always possible to find such an $L$ satisfiying $CG$ or $LG$.