@article{Brasco2022, 
author = {Lorenzo Brasco},
title = {Convex duality for principal frequencies},
year = {2022},
journal = {Mathematics in Engineering},
volume = {4},
number = {4},
pages = {1-28},
keywords = {torsional rigidity, Laplacian eigenvalues, inradius, Cheeger constant, geometric estimates, convex duality, hidden convexity},
url = {https://www.sciopen.com/article/10.3934/mine.2022032},
doi = {10.3934/mine.2022032},
abstract = {We consider the sharp Sobolev-Poincaré constant for the embedding of        W    0          1      ,      2        (  Ω  ) into        L    q    (  Ω  ). 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).}
}