In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, direct methods of the calculus of variations. A very closely related result is the Friedrichs' inequality.
Contents
The classical Poincaré inequality
Let p, so that 1 ≤ p < ∞ and Ω a subset with at least one bound. Then there exists a constant C, depending only on Ω and p , so that, for every function u of the W01,p(Ω) Sobolev space ,
The Poincaré-Wirtinger inequality
Assume that 1 ≤ p ≤ ∞ and that Ω is a bounded connected open subset of the n-dimensional Euclidean space Rn with a Lipschitz boundary (i.e., Ω is a Lipschitz domain). Then there exists a constant C, depending only on Ω and p, such that for every function u in the Sobolev space W1,p(Ω),
where
is the average value of u over Ω, with |Ω| standing for the Lebesgue measure of the domain Ω. When Ω is a ball, the above inequality is called a (p,p)-Poincaré inequality; for more general domains Ω, the above is more familiarly known as a Sobolev inequality.
Generalizations
In the context of metric measure spaces (for example, sub-Riemannian manifolds), such spaces support a (q,p)-Poincare inequality for some
In the context of metric measure spaces,
There exist other generalizations of the Poincaré inequality to other Sobolev spaces. For example, the following (taken from Garroni & Müller (2005)) is a Poincaré inequality for the Sobolev space H1/2(T2), i.e. the space of functions u in the L2 space of the unit torus T2 with Fourier transform û satisfying
there exists a constant C such that, for every u ∈ H1/2(T2) with u identically zero on an open set E ⊆ T2,
where cap(E × {0}) denotes the harmonic capacity of E × {0} when thought of as a subset of R3.
The Poincaré constant
The optimal constant C in the Poincaré inequality is sometimes known as the Poincaré constant for the domain Ω. Determining the Poincaré constant is, in general, a very hard task that depends upon the value of p and the geometry of the domain Ω. Certain special cases are tractable, however. For example, if Ω is a bounded, convex, Lipschitz domain with diameter d, then the Poincaré constant is at most d/2 for p = 1,
However, in some special cases the constant C can be determined concretely. For example, for p = 2, it is well known that over the domain of unit isosceles right triangle, C = 1/π ( < d/π where
Furthermore, for a smooth, bounded domain
and furthermore, that the constant λ1 is optimal.