Abstract

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.

Author information

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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

Keywords

  • Poincaré-Sobolev constant
  • p-Laplacian
  • inradius
  • distance function

Mathematics Subject Classification

46E35, 35J92, 35P30

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Suggested citation

F. Prinari, A. C. Zagati. “On the Sharp Makai Inequality.” Journal of Convex Analysis 31 (2024), No. 2, 709–732. https://doi.org/10.68381/jca31034

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Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.