Poincaré Type Inequalities

  • David R. Adams
  • Lars Inge Hedberg
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 314)


The term “Poincaré type inequality” is used, somewhat loosely, to describe a class of inequalities that generalize the classical Poincaré inequality,
$${\int_\Omega {\left| f \right|} ^p}dx \leqslant {A_\Omega }{\int_\Omega {\left| {\nabla f} \right|} ^p}dx$$
valid for fW 0 1,p (Ω) in a bounded open ΩR N . What the inequalities have in common is that an integral norm of a function is estimated in terms of integrals of its derivatives, and some information about the vanishing or the average of the function. Some such knowledge is clearly necessary, since estimates of this kind are false for non-zero constants.


Type Inequality Representation Formula Abstract Approach Basic Inequality Poincare Inequality 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • David R. Adams
    • 1
  • Lars Inge Hedberg
    • 2
  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA
  2. 2.Department of MathematicsLinköping UniversityLinköpingSweden

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