The study of the optimal constant 𝒦q(Ω) in the Sobolev inequality ||u||Lq(Ω)≤1/𝒦q(Ω)||Du||(ℝn), 1≤q<1*, for BV functions which are zero outside and with zero mean value inside , leads to the definition of a Cheeger type constant. We are interested in finding the best possible embedding constant in terms of the measure of alone. We set up an optimal shape problem and we completely characterize, on varying the exponent , the behaviour of optimal domains. Among other things we establish the existence of a threshold value 1≤q̅<1* above which the symmetry of optimal domains is broken. Several differences between the cases and are emphasized.
DOI : 10.1051/cocv/2014016
Mots-clés : Cheeger inequality, optimal shape, symmetry and asymmetry
@article{COCV_2015__21_2_359_0,
author = {Brandolini, Barbara and Della Pietra, Francesco and Nitsch, Carlo and Trombetti, Cristina},
title = {Symmetry breaking in a constrained {Cheeger} type isoperimetric inequality},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {359--371},
publisher = {EDP-Sciences},
volume = {21},
number = {2},
year = {2015},
doi = {10.1051/cocv/2014016},
mrnumber = {3348402},
zbl = {1319.49066},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2014016/}
}
TY - JOUR AU - Brandolini, Barbara AU - Della Pietra, Francesco AU - Nitsch, Carlo AU - Trombetti, Cristina TI - Symmetry breaking in a constrained Cheeger type isoperimetric inequality JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 359 EP - 371 VL - 21 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014016/ DO - 10.1051/cocv/2014016 LA - en ID - COCV_2015__21_2_359_0 ER -
%0 Journal Article %A Brandolini, Barbara %A Della Pietra, Francesco %A Nitsch, Carlo %A Trombetti, Cristina %T Symmetry breaking in a constrained Cheeger type isoperimetric inequality %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 359-371 %V 21 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014016/ %R 10.1051/cocv/2014016 %G en %F COCV_2015__21_2_359_0
Brandolini, Barbara; Della Pietra, Francesco; Nitsch, Carlo; Trombetti, Cristina. Symmetry breaking in a constrained Cheeger type isoperimetric inequality. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 359-371. doi : 10.1051/cocv/2014016. http://archive.numdam.org/articles/10.1051/cocv/2014016/
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