Ground state energy for a penetrable sphere and for a dielectric ball

We analyze the ultraviolet divergences in the ground state energy for a penetrable sphere and a dielectric ball. We argue that for massless fields subtraction of the ``empty space'' or the ``unbounded medium'' contribution is not enough to make the ground state energy finite whenever the heat kernel coefficient a₂ is not zero. It turns out that a₂≠0 for a penetrable sphere, a general dielectric background, and the dielectric ball. To our surprise, for more singular configurations, as in the presence of sharp boundaries, the heat kernel coefficients behave to some extent better than in the corresponding smooth cases, making, for instance, the dilute dielectric ball a well-defined problem.