Volume 25, 2019
|Number of page(s)||37|
|Published online||05 August 2019|
Interfacial energy as a selection mechanism for minimizing gradient Young measures in a one-dimensional model problem
Mathematical Institute, University of Oxford,
* Corresponding author: firstname.lastname@example.org
Accepted: 10 September 2018
Energy functionals describing phase transitions in crystalline solids are often non-quasiconvex and minimizers might therefore not exist. On the other hand, there might be infinitely many gradient Young measures, modelling microstructures, generated by minimizing sequences, and it is an open problem how to select the physical ones.
In this work we consider the problem of selecting minimizing sequences for a one-dimensional three-well problem ε. We introduce a regularization εε of ε with an ε-small penalization of the second derivatives, and we obtain as ε ↓ 0 its Γ-limit and, under some further assumptions, the Γ-limit of a suitably rescaled version of εε. The latter selects a unique minimizing gradient Young measure of the former, which is supported just in two wells and not in three. We then show that some assumptions are necessary to derive the Γ-limit of the rescaled functional, but not to prove that minimizers of εε generate, as ε ↓ 0, Young measures supported just in two wells and not in three.
Mathematics Subject Classification: 35B25 / 35B40 / 35Q74 / 49J45 / 74N15
Key words: Vanishing interface energy / selection mechanism / Young measures / three-well problem; Γ-limit
© EDP Sciences, SMAI 2019
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