Bhattacharya, Kaushik and Dolzmann, Georg (2001) Relaxation of some multiwell problems. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 131 (2). pp. 279320. ISSN 03082105. http://resolver.caltech.edu/CaltechAUTHORS:20131022085750715

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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 multiwell structure, i.e. W = 0 on a multiwell 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 twodimensional 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.
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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 MaxPlanckInstitute 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 496029810433).  
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Record Number:  CaltechAUTHORS:20131022085750715  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20131022085750715  
Official Citation:  Relaxation of some multiwell problems Proceedings Section A: Mathematics  Royal Society of Edinburgh, Volume 131, Number 2, 27 April 2001 , pp. 279320(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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