Symmetry breaking in a constrained Cheeger type isoperimetric inequality

The study of the optimal constant q(Ω) in the Sobolev inequality ||u||Lq(Ω)≤1/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