We analyze the ultraviolet divergences in the ground state energy for a pen
etrable sphere and a dielectric ball. We argue that for massless fields sub
traction of the ''empty space" or the "unbounded medium" contribution is no
t enough to make the ground state energy finite whenever the bent kernel co
efficient a(2) is not zero. It turns out that a(2) not equal 0 for a penetr
able sphere, a general dielectric background, and the dielectric ball. To o
ur 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 dielect
ric ball a well-defined problem. [S0556-2821(99)03508-0].