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 $$ of ε with an ε-small penalization of the second derivatives, and we obtain as ε ↓ 0 its Γ-limit and, under some further assumptions, the Γ-limit of a suitably rescaled version of $$. 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 $$ generate, as ε ↓ 0, Young measures supported just in two wells and not in three.
Accepté le :
DOI : 10.1051/cocv/2018050
Mots-clés : Vanishing interface energy, selection mechanism, Young measures, three-well problem, Γ-limit
@article{COCV_2019__25__A26_0,
author = {Della Porta, Francesco},
title = {Interfacial energy as a selection mechanism for minimizing gradient {Young} measures in a one-dimensional model problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018050},
zbl = {1445.35024},
mrnumber = {3989207},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018050/}
}
TY - JOUR AU - Della Porta, Francesco TI - Interfacial energy as a selection mechanism for minimizing gradient Young measures in a one-dimensional model problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018050/ DO - 10.1051/cocv/2018050 LA - en ID - COCV_2019__25__A26_0 ER -
%0 Journal Article %A Della Porta, Francesco %T Interfacial energy as a selection mechanism for minimizing gradient Young measures in a one-dimensional model problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018050/ %R 10.1051/cocv/2018050 %G en %F COCV_2019__25__A26_0
Della Porta, Francesco. Interfacial energy as a selection mechanism for minimizing gradient Young measures in a one-dimensional model problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 26. doi : 10.1051/cocv/2018050. http://www.numdam.org/articles/10.1051/cocv/2018050/
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