Relaxation of three solenoidal wells and characterization of three phase H-measures

We study the problem of characterizing quasiconvex hulls for three “solenoidal” (divergence free) wells in dimension three when the wells are pairwise incompatible. A full characterization for a generic regime is achieved by translating the problem into the language of H-measures, following recipes of Kohn and Smyshlyaev & Willis, in combination with certain ideas based on Šverák’s example of a “nontrivial” quasiconvex function. As a by-product, we obtain a new more general “geometrical” result: characterization of extremal three-point H-measures for three phase mixtures of characteristic functions in dimension three.