On the relation of a three-well energy

This paper studies the relaxation of ‘multi–well’ non–convex energies in the context of infinitesimal (‘geometrically linear’) elasticity (and the analogous problem for gradient fields). A ‘dual’ variational principle of Hashin–Shtrikman type is developed for the relaxed energy under the assumption that the original energy function is a difference of a quadratic function and of another convex function. In the particular case of linear elastic wells sharing the same elastic modulus, this reduces to the approach of R. V. Kohn and leads to the need to minimize an energy functional with respect to matrix measures on the unit sphere (the H–measures), which are subject to certain constraints. It is shown that for the ‘three–well’ linearized energy the minimizing measure subject to these constraints can be chosen as a sum of no more than three Dirac masses. A subclass of such three–point measures realizable by microstructures is described. It is demonstrated by means of examples that the minimizing measures do fall within this subclass in some cases, thereby providing an exact value for the three–well non–convex relaxed energy. More generally, the minimization algorithm leads to an ‘improved’ lower bound for the relaxed energy. If the three phases are pairwise compatible, the relaxed energy is a convexification of the original energy.