  The local Casimir energy density and the global Casimir energy for a massless
scalar field associated with a $\lambda\delta$-function potential in a 3+1
dimensional circular cylindrical geometry are considered. The global energy is
examined for both weak and strong coupling, the latter being the well-studied
Dirichlet cylinder case. For weak-coupling,through $\mathcal{O}(\lambda^2)$,
the total energy is shown to vanish by both analytic and numerical arguments,
based both on Green's-function and zeta-function techniques. Divergences
occurring in the calculation are shown to be absorbable by renormalization of
physical parameters of the model. The global energy may be obtained by
integrating the local energy density only when the latter is supplemented by an
energy term residing precisely on the surface of the cylinder. The latter is
identified as the integrated local energy density of the cylindrical shell when
the latter is physically expanded to have finite thickness. Inside and outside
the delta-function shell, the local energy density diverges as the surface of
the shell is approached; the divergence is weakest when the conformal stress
tensor is used to define the energy density. A real global divergence first
occurs in $\mathcal{O}(\lambda^3)$, as anticipated, but the proof is supplied
here for the first time; this divergence is entirely associated with the
surface energy, and does {\em not} reflect divergences in the local energy
density as the surface is approached.
