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Published April 2001 | Published
Journal Article Open

Relaxation of some multi-well problems


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.

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).

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