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Relaxation of three solenoidal wells and characterization of three-phase $H$-measures
Mariapia Palombaro and Valery P. Smyshlyaev
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 is achieved by combining certain ideas based on Šverák's example of a "nontrivial" quasiconvex function and on Müller's wavelet expansions estimates in terms of the Riesz transform. As a by-product, we obtain a new more general "geometrical" result: characterization of extremal three-point $H$-measures for three-phase mixtures in dimension three. We also discuss the applicability of the latter result to problems with other kinematic constrains, in particular to that of three linear elastic wells.