Relaxation of some multi-well problems
Kaushik Bhattacharya and Georg Dolzmann
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Submission date: 15. Nov. 1998
published in: Proceedings of the Royal Society of Edinburgh / A, 131 (2001) 2, p. 279-320
DOI number (of the published article): 10.1017/S0308210500000883
MSC-Numbers: 49J40, 52A30, 73B99, 73C50, 73V25
Keywords and phrases: nonconvex variational problems, generalized convex hulls, existence of minimizers, in-approximation, relaxed energy
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Mathematical models of phase transitions in solids lead to the variational problem, minimize where W has a multi-well structure: 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 or for with , in three dimensions when the matrices are essentially two-dimensional and also for for with which arises in the study of thin films. Here denotes the -matrix formed with the first two columns of . 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.