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MiS Preprint

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.

MSC Codes:
34A60, 49J45
differential inclusions, relaxation, $H$-measures and their characterization, Three-well problem, quasiconvex hulls, H-measures and their characterization

Related publications

2009 Journal Open Access
Mariapia Palombaro and Valery P. Smyshlyaev

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

In: Archive for rational mechanics and analysis, 194 (2009) 3, pp. 775-822