  The buckling instabilities of core-shell systems, comprising an interior
elastic sphere, attached to an exterior shell, have been proposed to underlie
myriad biological morphologies. To fully discuss such systems, however, it is
important to properly understand the elasticity of the spherical core. Here, by
exploiting well-known properties of the solid harmonics, we present a simple,
direct method for solving the linear elastic problem of spheres and spherical
voids with surface deformations, described by a real spherical harmonic. We
calculate the corresponding bulk elastic energies, providing closed-form
expressions for any values of the spherical harmonic degree (l), Poisson ratio,
and shear modulus. We find that the elastic energies are independent of the
spherical harmonic index (m). Using these results, we revisit the buckling
instability experienced by a core-shell system comprising an elastic sphere,
attached within a membrane of fixed area, that occurs when the area of the
membrane sufficiently exceeds the area of the unstrained sphere [C. Fogle, A.
C. Rowat, A. J. Levine and J. Rudnick, Phys. Rev. E 88, 052404 (2013)]. We
determine the phase diagram of the core-shell sphere's shape, specifying what
value of l is realized as a function of the area mismatch and the core-shell
elasticity. We also determine the shape phase diagram for a spherical void
bounded by a fixed-area membrane.
