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The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3522–3552 | Cite as

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

  • Andrea Cianchi
  • Vincenzo Ferone
  • Carlo Nitsch
  • Cristina TrombettiEmail author
Article
  • 80 Downloads

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 \(\Omega \).

Keywords

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

Mathematics Subject Classification

46E35 26B30 

Notes

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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Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  • Andrea Cianchi
    • 1
  • Vincenzo Ferone
    • 2
  • Carlo Nitsch
    • 3
  • Cristina Trombetti
    • 3
    Email author
  1. 1.Dipartimento di Matematica e Informatica “U. Dini”Università di FirenzeFirenzeItaly
  2. 2.Dipartimento di Fisica “E. Pancini”Università di Napoli “Federico II”NapoliItaly
  3. 3.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”Università di Napoli “Federico II”NapoliItaly

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