Preprint 50/1998

Relaxation of some multi-well problems

Kaushik Bhattacharya and Georg Dolzmann

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Submission date: 15. Nov. 1998
Pages: 39
published in: Proceedings of the Royal Edinburgh Society / 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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Abstract:
Mathematical models of phase transitions in solids lead to the variational problem, minimize tex2html_wrap_inline17 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 tex2html_wrap_inline27 or tex2html_wrap_inline29 for tex2html_wrap_inline31 with tex2html_wrap_inline33, in three dimensions when the matrices tex2html_wrap_inline35 are essentially two-dimensional and also for tex2html_wrap_inline37 for tex2html_wrap_inline39 with tex2html_wrap_inline41 which arises in the study of thin films. Here tex2html_wrap_inline43 denotes the tex2html_wrap_inline45-matrix formed with the first two columns of tex2html_wrap_inline35. 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.

21.02.2013, 01:40