Symmetry breaking in a constrained cheeger type isoperimetric inequality

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

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


The study of the optimal constant Kq(Ω) in the Sobolev inequality ∥u∥Lq(Ω) ≤ 1/Kq(Ω)∥Du∥(double-struck 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 ≤ q < 1∗ above which the symmetry of optimal domains is broken. Several differences between the cases n = 2 and n ≥ 3 are emphasized.
Original languageEnglish
Pages (from-to)359-371
Number of pages13
Publication statusPublished - 2015

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Control and Optimization
  • Computational Mathematics


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