Symmetry breaking in a constrained Cheeger type isoperimetric inequality

The study of the optimal constant $mathcal{K}_q(Omega)$ in the Sobolev inequality $$ ||u||_{L^q(Omega)}le rac{1}{mathcal{K}_q(Omega)}||Du||(mathbb{R}^n),qquad 1 le q <1^ast, $$ for BV functions which are zero outside $Omega$ and with zero mean value inside $Omega$, 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 $Omega$ 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 $1le ilde{q}<1^ast$ above which the symmetry of optimal domains is broken. Several differences between the cases $n = 2$ and $n ge 3$ are emphasized.