We consider the minimization of averaged shape optimization problems over the class of sets of finite perimeter. We use occupational measures, which are probability measures defined in terms of the reduced boundary of sets of finite perimeter, that allow to transform the minimization into a linear problem on a set of measures. The averaged nature of the problem allows the optimal value to be approximated with sets with unbounded perimeter. In this case, we show that we can also approximate the optimal value with convex polytopes with n+1 faces shrinking to a point. We derive conditions under which we show the existence of minimizers and we also analyze the appropriate spaces in which to study the problem.
Accepté le :
DOI : 10.1051/cocv/2017017
Mots clés : Shape optimization, occupational measures, sets of finite perimeter, Cheeger sets
@article{COCV_2018__24_3_1141_0,
author = {Bright, Ido and Li, Qinfeng and Torres, Monica},
title = {Occupational measures and averaged shape optimization},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1141--1165},
publisher = {EDP-Sciences},
volume = {24},
number = {3},
year = {2018},
doi = {10.1051/cocv/2017017},
zbl = {1405.49030},
mrnumber = {3877196},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2017017/}
}
TY - JOUR AU - Bright, Ido AU - Li, Qinfeng AU - Torres, Monica TI - Occupational measures and averaged shape optimization JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1141 EP - 1165 VL - 24 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2017017/ DO - 10.1051/cocv/2017017 LA - en ID - COCV_2018__24_3_1141_0 ER -
%0 Journal Article %A Bright, Ido %A Li, Qinfeng %A Torres, Monica %T Occupational measures and averaged shape optimization %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1141-1165 %V 24 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2017017/ %R 10.1051/cocv/2017017 %G en %F COCV_2018__24_3_1141_0
Bright, Ido; Li, Qinfeng; Torres, Monica. Occupational measures and averaged shape optimization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1141-1165. doi : 10.1051/cocv/2017017. http://www.numdam.org/articles/10.1051/cocv/2017017/
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