On a class of Cheeger inequalities

Abstract

We study a general version of the Cheeger inequality by considering the shape functional Fp,q(Ω)=λ1/pp(Ω)/λ1/qq(Ω). The infimum and the supremum of Fp,q are studied in the class of all domains Ω of Rd and in the subclass of convex domains. In the latter case the issue concerning the existence of an optimal domain for Fp,q is discussed.