Issue |
ESAIM: COCV
Volume 21, Number 2, April-June 2015
|
|
---|---|---|
Page(s) | 359 - 371 | |
DOI | https://doi.org/10.1051/cocv/2014016 | |
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
Cintia, 80126
Napoli,
Italia.
brandolini@unina.it; f.dellapietra@unina.it;
c.nitsch@unina.it; cristina@unina.it
Received:
7
November
2013
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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