Poincaré Trace Inequalities in \(\textit{BV}({\mathbb {B}}^n)\) with Non-standard Normalization

Abstract

Extremal functions are exhibited in Poincaré trace inequalities for functions of bounded variation in the unit ball \({\mathbb {B}}^n\) of the n-dimensional Euclidean space \({{\mathbb {R}}}^n\). Trial functions are subject to either a vanishing mean value condition, or a vanishing median condition in the whole of \({\mathbb {B}}^n\), instead of just on \(\partial {\mathbb {B}}^n\), as customary. The extremals in question take a different form, depending on the constraint imposed. In particular, under the median constraint, unusually shaped extremal functions appear. A key step in our approach is a characterization of the sharp constant in the relevant trace inequalities in any admissible domain \(\Omega \subset {{\mathbb {R}}}^n\), in terms of an isoperimetric inequality for subsets of \(\Omega \).

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Acknowledgements

This research was partly supported by the research project of MIUR (Italian Ministry of Education, University and Research) Prin 2012, No. 2012TC7588, “Elliptic and parabolic partial differential equations: geometric aspects, related inequalities, and applications,” and by GNAMPA of the Italian INdAM (National Institute of High Mathematics).

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Correspondence to Cristina Trombetti.

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Cianchi, A., Ferone, V., Nitsch, C. et al. Poincaré Trace Inequalities in \(\textit{BV}({\mathbb {B}}^n)\) with Non-standard Normalization. J Geom Anal 28, 3522–3552 (2018). https://doi.org/10.1007/s12220-017-9968-z

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Keywords

  • Boundary traces
  • Sharp constants
  • Poincaré inequalities
  • Functions of bounded variation
  • Sobolev spaces
  • Isoperimetric inequalities

Mathematics Subject Classification

  • 46E35
  • 26B30