We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
111/2007
Relaxation of three solenoidal wells and characterization of three-phase $H$-measures
Mariapia Palombaro and Valery P. Smyshlyaev
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 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.
differential inclusions, relaxation, $H$-measures and their characterization, Three-well problem, quasiconvex hulls, H-measures and their characterization
Related publications
inJournal
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
