Symmetry breaking in a constrained Cheeger type isoperimetric inequality

The study of the optimal constant $\mathcal K_q(\Omega)$ in the Sobolev inequality \[ \|u\|_{L^q(\Omega)} \le \frac{1}{\mathcal K_q(\Omega)}\|Du \|(\R^n), \qquad 1\le q<1^\ast, \] 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 $1\le\tilde q< 1^\ast$ above which the symmetry of optimal domains is broken. Several differences between the cases $n=2$ and $n\ge 3$ are emphasized.