Minimal Energy for Geometrically Nonlinear Elastic Inclusions in Two Dimensions

We are concerned with 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. Following ideas from \cite{CS} and \cite{KnuepferKohn-2011}, 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.