If one models the electron as a hollow spherical conductor with charge $e$ and radius $a$ then its electrostatic energy is given by:

$$E_{em}=\frac{1}{2}\frac{e^2}{4\pi\epsilon_0a}$$

However if one calculates the momentum in the field of a moving electron then one finds that the total mass in the field is given by:

$$m_{em}=\frac{2}{3}\frac{e^2}{4\pi\epsilon_0c^2a}$$

Therefore we have a discrepancy in energy given by:

$$E_p = \frac{1}{6}\frac{e^2}{4\pi\epsilon_0a}$$

Poincare hypothesised that there must be stresses holding the electron together against the electrostatic repulsion of the charge on its surface. Somehow it must take an energy $E_p$ in order to maintain these Poincare stresses.

Perhaps the electron can be modelled by a charged spherical shell with a vacuum inside it. Presumably the vacuum would lead to a negative pressure on the charged shell due to the Casimir effect. This pressure must balance the electrostatic repulsion of the charged shell.

According to cosmological models the vacuum has an equation of state given by:

$$p = -\rho c^2$$

The outward pressure on the surface of the charged sphere due to its charge is given by:

$$p = \frac{1}{2}\epsilon_0\left(\frac{e}{4\pi\epsilon_0a^2}\right)^2$$

This pressure must be balanced by the negative pressure of the vacuum inside the shell. If we substitute into the above equation of state for a vacuum we find that its energy must be given by: $$E_p=\frac{1}{6}\frac{e^2}{4\pi\epsilon_0a}$$

Thus we seem to have accounted for the energy discrepancy between the EM field of a static electron and a moving electron by including the energy of the vacuum inside the electron holding it together.

Does this make sense?

P.S. The Compton wavelength of the electron, $\lambda = h / m_e c$, is approximately $10^{-12}\ $m. Thus the energy density of the vacuum inside the electron is:

$$\rho c^2 \approx 10^{31}\ \mbox{GeV/m^3}$$

This is vastly more than the vacuum energy density in currently accepted cosmological models (a few GeV/m^3). I guess a big difference is that the Universe is not bounded by a conductor as in the above model of an electron.