Conditions for applying Gauss' Law

[jv07cs]

To apply the Divergence Theorem (DT), at least as it is stated and proved in undergrad calculus, it is required for the vector field $\vec{F}$ to be defined both on the surface ∂V, so that we can evaluate the flux through this surface, and on the volume V enclosed by ∂V, so that we can evaluate the integral of $\nabla\cdot\vec{F}$ on V. In electrostatics, we use Gauss' Law (GL), which comes directly from the DT.

From my understanding, to apply Gauss' Law, we would then have to require for $\vec{E}$ to be defined both on the surface ∂V and on the volume V. If $\vec{E}$ is not defined on V, we can get around this by using the dirac delta function. But what if $\vec{E}$ is not defined on the boundary ∂V?

More specifically, I would like to ask about two examples. When we consider a uniformly charged sphere of radius $R$, for example, and we use Gauss' Law to calculate $\vec{E}$ inside the sphere, drawing a Gaussian Sphere $V_0$ with boundary $\partial V_0$ and radius $r < R$. To apply the theorem in the first place, we needed to know that $\vec{E}$ is indeed defined inside the sphere, which is not necessarily obvious from Coulomb's Law

as there are singularities since cursive r is on the denominator. So we would need to know that this integral converges. How can I be sure of this before applying Gauss' Law? Does it come from the fact that $\nabla\cdot\vec{E} (\vec{r}) = \rho(\vec{r})/\epsilon_0$? Since the divergence of $\vec{E}$ inside the distribution is defined, as $\rho(\vec{r})$ is defined, then $\vec{E}(\vec{r})$ must also be defined.

The other question is regarding the uniformly charged infinite plane, for example. In this case, we use a pillbox, with volume $V_0$ and boundary $\partial V_0$, as the gaussian surface and, applying Gauss' Law to calculate the field at a height $z$ and then taking the limit $z\to 0$, we find out that the field is not defined on the plane, there is a discontinuity. If the field is not defined on the plane and the pillbox pierces the plane, then the field is not defined everywhere on the surface of the pillbox, that is $\vec{E}$ is not defined on $\partial V_0$. From the requirement that the field is defined on $\partial V_0$ for the DT to be applied, we could not apply the DT here. Mathematically speaking, what allows us to apply Gauss' Law here?

 

[anuttarasammyak]

Does it come from the fact that ∇⋅E→(r→)=ρ(r→)/ϵ0?

I think that finite $\rho$ assures it.

The other question is regarding the uniformly charged infinite plane, for example.

E_z(z)=\frac{\sigma}{2\epsilon_0}sgn(z)
Ambiguity of
sgn(z):=2(H_a(z)-1/2)
, where usually a =1/2 but another value in general, at z=0 would not show difficulty in so much for physics.

[EDIT] Distribution of charge area density
\sigma(z)=\sigma \delta(z)
With a tricky relation
\int_0^{+\epsilon} \delta(z) =\frac{1}{2}
Applying Gauss's law for [0,+\epsilon] pillbox, we might be able to say
E_z(0)=0
which means a=1/2 as usual.