Convex duality for principal frequencies

TL;DR

This paper reveals a dual variational formulation for the Sobolev-Poincaré constant in the range 1<q<2, enabling new lower bound proofs and extending classical results related to torsional rigidity and eigenvalues.

Contribution

It introduces a convex minimization framework with a divergence constraint for the Sobolev-Poincaré constant, generalizing known cases for torsional rigidity and eigenvalues.

Findings

Dual variational formulation for 1<q<2.

Convex minimization problem with divergence constraint.

Extension of classical results to broader q-range.

Abstract

We consider the sharp Sobolev-Poincar\'e constant for the embedding of $W^{1,2}_0(\Omega)$ into $L^q(\Omega)$. 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 $q=2$).