Poincaré inequality


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 Friedrichs' inequality.

Statement of the inequality

The classical Poincaré inequality

Let p, so that 1 ≤ p < ∞ and Ω a subset bounded at least in one direction. Then there exists a constant C, depending only on Ω and p, so that, for every function u of the Sobolev space W01,p of zero-trace functions,

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. Then there exists a constant C, depending only on Ω and p, such that for every function u in the Sobolev space W1,p,
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 -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, such spaces support a -Poincare inequality for some if there are constants C and so that for each ball B in the space,
In the context of metric measure spaces, is the minimal p-weak upper gradient of u in the sense of
Heinonen and Koskela
There exist other generalizations of the Poincaré inequality to other Sobolev spaces. For example, the following is a Poincaré inequality for the Sobolev space H1/2, 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 with u identically zero on an open set E ⊆ T2,
where cap denotes the harmonic capacity of E × 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, for p = 2, and this is the best possible estimate on the Poincaré constant in terms of the diameter alone. For smooth functions, this can be understood as an application of the isoperimetric inequality to the function's level sets. In one dimension, this is Wirtinger's inequality for functions.
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.
Furthermore, for a smooth, bounded domain, since the Rayleigh quotient for the Laplace operator in the space is minimized by the eigenfunction corresponding to the minimal eigenvalue λ1 of the Laplacian, it is a simple consequence that, for any,
and furthermore, that the constant λ1 is optimal.