Distributional limits concerning the regularity of Maxwells equations

Question

This question is related to my previous question about the regularity of the Maxwell equations.

Assume we are working on a space where there are only electric point charges, $(q_i)$, and a blob of matter $M$ which is a bounded, closed and connected set in $\mathbf R^3$. With respect to the electric field, the only difference is the permittivity $\varepsilon$. That is, $\varepsilon$ is a function which is dependent on the space variables $(x, y, z)$. In this particular $M$ is homogeneous so $\varepsilon$ is constant almost everywhere except on the boundary of $M$, where it has a jump.

If we are given the potential $V$ of this system, then the electric field $\mathbf E = -\nabla V$ will likewise make a jump. My question is now related to expressions of the form. $$\int_{\mathbf R^3} |\mathbf E|^2 \ \nabla \varepsilon \ \mathrm{d}V.\tag{*}$$

Hence, to my understanding we have returned to the regularity of the solutions of the Maxwell equations. $\nabla \varepsilon$ will contain a $\delta$-distribution and in its support $\mathbf E$, $(*)$ is not so well-defined as it would seem. The result would strongly depend on the chosen approximating sequence to $\delta$. This was the problem and the Maxwell equations are one particular example where I have seen them occur.

But, of course, if we instead of having $\varepsilon$ have a jump, we could have a smooth -but fast- transition. I could have a sequence $(\varepsilon_n)_n$ of those functions that converge distributionally to the Heaviside distribution. However, $\varepsilon$ determines the solution $\mathbf E$ so, $\mathbf E$ will be different too.

My questions:

1. Can expressions of the form $(*)$ where we are actually evaluation a function on a null-set which is not well-defined on that point? In PDE we would use traces for this. In particular, we map the boundary to a space where it has positive measure and work there. Is there a general theme for such expression as the one here?
2. The expression above can have an approximation as $\nabla \varepsilon_n$ somehow. We know that $\mathbf E$ is the solution for the limit $\lim_n \nabla \varepsilon_n$, can we somehow reduce the question to computing (for instance) $$\lim_{n \to \infty} \int_{\mathbf R^3} |\mathbf E|^2 \ \nabla \varepsilon_n \ \mathrm{d}V,$$ or will this never be well-defined? The problem here of course is that $\varepsilon_n$ is not the "correct" one that matches $\mathbf E$.
3. How is the question related to regularity? Naively I would expect that even though the distribution $(*)$ is not well-defined as an integral, that $\lim_n |\mathbf E_n|^2 \nabla \varepsilon_n = |\mathbf E|^2 \nabla \varepsilon$ distributionally.
4. In hyperbolic equations, if we change some of the parameters locally, does the global solution "feel" this? If a wave is propagating and I change some of the material it will go through way before the wave arrived there, the current solution should not change. Is this a general property? (Err... I realise this is basically the definition, but the question is thus is if this makes the above property easier to prove. I would suspect so.).

And, as before, I know little to nothing about hyperbolic equations. As per suggestion the last time, I do have the book "Multi-dimensional hyperbolic partial differential equations" by Sylvie Benzoni-Gavage and Denis Serre (but so much material and takes some time to digest). However, I do know a bit about the elliptic theory and now also the Strichartz estimates.

Answer 1

Yes, there is a way of interpreting this integral. Decompose E into a component $E_t$ tangential to level sets of $\epsilon$ and a component $E_n$ normal to them. Moreover, let us set $D=\epsilon E$, and decompose it in the same way. You can write your integrand as $E_t^2\nabla \epsilon-D_n^2\nabla(1/\epsilon)$. Now if you take the limit of a discontinuous material boundary, then both $E_t$ and $D_n$ are continuous, so there is no problem definining the integral in this form.