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Isoperimetric inequalities for the principal eigenvalues of the Robin-Laplacian are interpreted as free discontinuity problems (of unusual type). We prove a full range of Faber–Krahn inequalities in a nonlinear setting and for non smooth domains, including the open case of the torsional rigidity. The key point of the analysis relies on regularity issues for free discontinuity problems in spaces of functions of bounded variation. As a byproduct, we obtain the best constants for a class of Poincaré inequalities with trace terms in .