Poincaré Inequality

The Poincaré inequality is an estimate for functions on a bounded domain in terms of their gradients and averages. Let be an open, bounded, and connected subset of with Lipschitz boundary, for some and let denote -dimensional Lebesgue measure on . In functional analysis, the Poincaré inequality says that there exist constants and such that

for all real-valued functions in the Sobolev space consisting of all functions in whose weak first derivatives are all also square integrable.

This inequality plays an important role in the study of both function spaces and partial differential equations. As such, a number of generalizations have been established to domains and functions which are less well-behaved, e.g., to polyhedral domains and to functions which only have desirable behavior piecewise on .

In some literature, the above-stated Poincaré inequality is sometimes referred to as the mean Poincaré inequality with the unqualified phrase "Poincaré inequality" reserved for the so-called (and closely-related) Friedrichs inequality. Inequalities which are qualitatively similar to those of Friedrichs and Poincaré are sometimes referred to collectively as Poincaré-Friedrichs inequalities.

For a probability measure , the least constant in the variance form of the inequality is its Poincaré constant . Its dimension-independent boundedness for isotropic logarithmically concave distributions is the Kannan-Lovász-Simonovits conjecture.