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 geometrically nonlinear two-well problem elasticity. More precisely, we investigate optimal scaling energy an elastic inclusion fixed volume for is determined by surface and (anisotropic) contribution. Following ideas from Conti Schweizer ( Commun. Pure Appl. Math. 59 (2006), 830–868) Knüpfer Kohn Proc. R. Soc. London Ser. A Phys. Eng. Sci. 467 (2011), 695–717), derive lower bound invoking rigidity argument covering result. The upper follows well-known construction lens-shaped inclusion.