Journal of Convex Analysis 31 (2024), No. 2, 709--732
Copyright Heldermann Verlag 2024
On the Sharp Makai Inequality
Francesca Prinari
Dip. di Scienze Agrarie, Alimentari e Agro-Ambientali, Università di Pisa, Pisa, Italy
francesca.prinari@unipi.it
Anna Chiara Zagati
Dip. di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parma, Italy
annachiara.zagati@unipr.it
On a convex bounded open set, we prove that Poincaré-Sobolev constants for functions vanishing at the boundary can be bounded from below in terms of the norm of the distance function in a suitable Lebesgue space. This generalizes a result shown, in the planar case, by E. Makai, for the torsional rigidity. In addition, we compare the sharp Makai constants obtained in the class of convex sets with the optimal constants defined in other classes of open sets. Finally, an alternative proof of the Hersch-Protter inequality for convex sets is given.
Keywords: Poincaré-Sobolev constant, p-Laplacian, inradius, distance function.
MSC: 46E35, 35J92, 35P30.
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