Relaxation of three solenoidal wells and characterization of three-phase H-measures
Mariapia Palombaro and Valery P. Smyshlyaev
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Submission date: 19. Dec. 2007
published in: Archive for rational mechanics and analysis, 194 (2009) 3, p. 775-822
DOI number (of the published article): 10.1007/s00205-008-0204-7
MSC-Numbers: 34A60, 49J45
Keywords and phrases: differential inclusions, relaxation, $H$-measures and their characterization, Three-well problem, quasiconvex hulls, H-measures and their characterization
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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 Sverák'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.