Preprint 111/2007

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
Pages: 44
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.

21.02.2013, 01:42