Volume 21, Number 2, April-June 2015
|Page(s)||359 - 371|
|Published online||10 October 2014|
Symmetry breaking in a constrained Cheeger type isoperimetric inequality
Universitàdegli Studi di Napoli
“Federico II”, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Complesso
Monte S. Angelo - Via
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Revised: 21 February 2014
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 n ≥ 3 are emphasized.
Mathematics Subject Classification: 49Q20 / 39B05
Key words: Cheeger inequality / optimal shape / symmetry and asymmetry
© EDP Sciences, SMAI 2014
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