Symmetry breaking in a constrained Cheeger type isoperimetric inequality
ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 2, pp. 359-371.

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 Ω and with zero mean value inside Ω, 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 Ω alone. We set up an optimal shape problem and we completely characterize, on varying the exponent q, 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 n=2 and n3 are emphasized.

Received:
DOI: 10.1051/cocv/2014016
Classification: 49Q20, 39B05
Keywords: Cheeger inequality, optimal shape, symmetry and asymmetry
Brandolini, Barbara 1; Della Pietra, Francesco 1; Nitsch, Carlo 1; Trombetti, Cristina 1

1 Universitàdegli Studi di Napoli “Federico II”, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Complesso Monte S. Angelo - Via Cintia, 80126 Napoli, Italia.
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     title = {Symmetry breaking in a constrained {Cheeger} type isoperimetric inequality},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {359--371},
     publisher = {EDP-Sciences},
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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, Volume 21 (2015) no. 2, pp. 359-371. doi : 10.1051/cocv/2014016. http://www.numdam.org/articles/10.1051/cocv/2014016/

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