Convex duality for principal frequencies

Lorenzo Brasco; Lorenzo Brasco

doi:10.3934/mine.2022032

Mathematics in Engineering

2022, Volume 4, Issue 4: 1-28. doi: 10.3934/mine.2022032

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Convex duality for principal frequencies

Lorenzo Brasco ^,

Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli 30, 44121 Ferrara, Italy

Received: 10 June 2021 Accepted: 30 August 2021 Published: 17 September 2021

We consider the sharp Sobolev-Poincaré 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 $).
- torsional rigidity,
- Laplacian eigenvalues,
- inradius,
- Cheeger constant,
- geometric estimates,
- convex duality,
- hidden convexity
Citation: Lorenzo Brasco. Convex duality for principal frequencies[J]. Mathematics in Engineering, 2022, 4(4): 1-28. doi: 10.3934/mine.2022032

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Abstract

We consider the sharp Sobolev-Poincaré 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 $).

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