

Preprint 111/2007
Relaxation of three solenoidal wells and characterization of three-phase H-measures
Mariapia Palombaro and Valery P. Smyshlyaev
Contact the author: Please use for correspondence this email.
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
Bibtex
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
Download full preprint: PDF (397 kB)
Abstract:
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.