Calculus of Variations and Geometric Measure Theory

L. Brasco

Convex duality for principal frequencies

created by brasco on 12 May 2021
modified on 30 Aug 2021


Accepted Paper

Inserted: 12 may 2021
Last Updated: 30 aug 2021

Journal: Math. Eng.
Pages: 25
Year: 2021


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 $q=2$).

Keywords: Torsional rigidity, Cheeger constant, Inradius, hidden convexity, convex duality, geometric estimates, Laplacian eigenvalues