Convex duality for principal frequencies. (English) Zbl 1496.35015
Summary: 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 \(q = 2\)).
MSC:
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35P15 | Estimates of eigenvalues in context of PDEs |
Keywords:
torsional rigidity; Laplacian eigenvalues; inradius; Cheeger constant; geometric estimates; convex duality; hidden convexityReferences:
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