We investigate a variant of the isoperimetric problem, which in our
setting arises in a geometrically nonlinear two-well problem in
elasticity. More precisely, we investigate the optimal scaling of the energy
of an elastic inclusion of a fixed volume for which the energy is determined
by a surface and an (anisotropic) elastic contribution. We derive the lower scaling bound by invoking a
two-well rigidity argument and a covering result. The upper bound follows from
a well-known construction for a lens-shaped elastic inclusion. This is joint work with I. Akramov, H. Knuepfer, and A. Rueland.