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Relaxation of some multi-well problems

Bhattacharya, Kaushik and Dolzmann, Georg (2001) Relaxation of some multi-well problems. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 131 (2). pp. 279-320. ISSN 0308-2105. http://resolver.caltech.edu/CaltechAUTHORS:20131022-085750715

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Abstract

Mathematical models of phase transitions in solids lead to the variational problem, minimize ∫_ΩW (Du) dx, where W has a multi-well structure, i.e. W = 0 on a multi-well set K and W > 0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e. for K = SO(2)U_1 ∪...∪ SO(2)U_k or K = O(2)U_1 ∪...∪ O(2)Uk for U_1,...,U_k ∈ M^(2×2) with det U_i = δ in three dimensions when the matrices U_i are essentially two-dimensional and also for K = SO(3)Û_1 ∪...∪ SO(3)Û_k for U_1,...,U_k ∈ M^(3×3) with (adj U_i^TU_i)33 = δ^2, which arises in the study of thin films. Here, Û_i denotes the (3×2) matrix formed with the first two columns of U_i. We characterize generalized convex hulls, including the quasiconvex hull, of these sets, prove existence of minimizers and identify conditions for the uniqueness of the minimizing Young measure. Finally, we use the characterization of the quasiconvex hull to propose 'approximate relaxed energies', quasiconvex functions which vanish on the quasiconvex hull of K and grow quadratically away from it.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1017/S0308210500000883DOIArticle
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1201332PublisherArticle
ORCID:
AuthorORCID
Bhattacharya, Kaushik0000-0003-2908-5469
Additional Information:© 2001 Royal Society of Edinburgh. Received 21 October 1998; accepted 11 May 1999. This work was conducted while K.B. visited the MPI Leipzig in 1997. 199 and while G.D. held a position at the Max-Planck-Institute for Mathematics in the Sciences. Leipzig, and also visited Caltech in 1998. We gratefully acknowledge the partial financial support of the NSF (CMS 9457573) and AFOSR/MURI (F 49602-98-1-0433).
Funders:
Funding AgencyGrant Number
NSFCMS 9457573
Air Force Office of Scientific Research (AFOSR)F 49602-98-1-0433
Record Number:CaltechAUTHORS:20131022-085750715
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20131022-085750715
Official Citation:Relaxation of some multi-well problems Proceedings Section A: Mathematics - Royal Society of Edinburgh, Volume 131, Number 2, 27 April 2001 , pp. 279-320(42) Authors: Bhattacharya, Kaushik; Dolzmann, Georg
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:42004
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:22 Oct 2013 19:57
Last Modified:26 Aug 2015 23:21

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