Minimal energy for elastic inclusions

We consider a variant of the isoperimetric problem with a non-local term representingelastic energy. More precisely, our aim is to analyse the optimal energy of an inclusion of a fixed volume the energy of which is determined by surface and elastic energies. This problem has been studied extensively in the physical/metallurgical literature; however, the analysis has mainly been either (i) numerical, or (ii) restricted to a specific set of inclusion shapes, e.g. ellipsoids. In this article, we prove a lower bound for the energy, with no a priori hypothesis on the shape (or even number) of the inclusions. This journal is