We consider the sharp Sobolev-Poincaré constant for the embedding of W01,2(Ω) into Lq(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).
Convex duality for principal frequencies
Brasco L.
Primo
2022
Abstract
We consider the sharp Sobolev-Poincaré constant for the embedding of W01,2(Ω) into Lq(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).File in questo prodotto:
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