Abstract
<abstract><p>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 $).</p></abstract>
Publisher
American Institute of Mathematical Sciences (AIMS)
Subject
Applied Mathematics,Mathematical Physics,Analysis
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