Symmetry breaking in a constrained Cheeger type isoperimetric inequality

Brandolini, Barbara; Della Pietra, Francesco; Nitsch, Carlo; Trombetti, Cristina

doi:10.1051/cocv/2014016

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.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 359-371.

Résumé

The study of the optimal constant 𝒦_q(Ω) in the Sobolev inequality ||u||_L^q(Ω)≤1/𝒦_q(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 $n \geq 3$ are emphasized.

Reçu le : 2013-11-07

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/cocv/2014016

Classification : 49Q20, 39B05
Mots clés : Cheeger inequality, optimal shape, symmetry and asymmetry

Affiliations des auteurs :

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},
     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/}
}

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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, 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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