Convex duality for principal frequencies

<abstract><p>We consider the sharp Sobolev-Poincaré constant for embedding of $ W^{1, 2}_0(\Omega) into L^q(\Omega) $. We show that such a exhibits an unexpected dual variational formulation, in range 1 < q 2 Namely, this can be written as convex minimization problem, under divergence–type constraint. This is particularly useful order to prove lower bounds. The result generalizes what happens torsional rigidity (corresponding = $) and extends up case first eigenvalue Dirichlet-Laplacian (i.e., $).</p></abstract>