Convex duality for principal frequencies

We consider the sharp Sobolev-Poincaré constant for the embedding of $W_0^{1,2}(\mathrm{\Omega })$ into $L^q(\mathrm{\Omega })$. We show that such a constant exhibits an unexpected dual variational formulation, in the range $1<q<2$. Namely, this can be written as a convex minimization problem, under a divergence–type constraint. This is particularly useful in order to prove lower bounds. The result generalizes what happens for the torsional rigidity (corresponding to $q=1$) and extends up to the case of the first eigenvalue of the Dirichlet-Laplacian (i.e., to $q=2$).