Poincaré inequality

In calculus is called the Poincaré inequality a named after the French mathematician Henri Poincaré inequality from the theory of Sobolev spaces. The inequality makes it possible to derive bounds on the barrier function of the derivatives, and the geometry of the domain. Such barriers play a major role in the calculus of variations.

Formulation of the inequality

The classical Poincaré inequality

Let 1 ≤ p ≤ ∞ and Ω a bounded connected open subset of n-dimensional Euclidean space Rn with Lipschitz boundary (ie, Ω is a Lipschitz region). Then there is a constant C that depends only on Ω and p, such that for each u in function Sobolev space W1, P ( Ω ), the inequality

Holds, where

The mean value of u on Ω is | Ω | denotes the Lebesgue measure of the region Ω.

With the help of Hölder 's inequality, one can show that the Poincaré inequality follows from the Poincaré inequality. General: if the Poincaré inequality for some p for a region Ω ' is true, then it also holds for all p > p', possibly with a different constant C.

One -dimensional example

Let f be a continuously differentiable function with Fourier series

Then using the Parseval equation

Manifolds

For Riemannian manifolds with nonnegative Ricci curvature (for example, non-negative sectional curvature ), the Poincaré inequality holds. There is only a constant of the dimension n abhängende, such that for all:

Metric spaces

Bruce Kleiner proved 2007, a Poincaré inequality for the Cayley graphs of finitely generated groups:

With a piecewise smooth function, its average value over the ball and the Cayley graph defining generating system is. Using this inequality, he gave a simplified proof of theorem on groups of polynomial growth Gromows.

For metric spaces of non- negative Ricci curvature in the sense of Lott - Villani - Sturm the weak local Poincaré inequality was proved in 2012 by Rajala.

Generalizations

There are generalizations of the Poincaré inequality for other Sobolev spaces. For example, the following inequality for the Poincaré Sobolev space H1 / 2 (T2), that is the space of functions u in L2 space of the torus T2 whose Fourier transform û the condition

True: There is a constant C such that for every u ∈ H1 / 2 (T2) with identical u 0 on an open set E ⊆ T2, the following inequality holds:

Said cap (E × {0} ) indicates the harmonic capacity of E × {0} as a subset of R3.

The Poincaré constant

The optimal constant C in the Poincaré inequality is called the Poincaré constant of the region Ω. It is generally very hard to determine the Poincaré constant, depending on p and the geometry of the area Ω. Certain special cases are but treatable. For example, for a limited, convex areas Lipschitz Ω with diameter d the Poincaré constant is at most D / 2, if p = 1, and at most d / π when p = 2, and this is the best possible estimate of the diameter of only abhängende the Poincaré constant. For obtaining the smooth functions as an application of the inequality isoperimetric level amounts of the function. In the one -dimensional that is the Wirtinger 's inequality for functions.

There are special cases where the constant C can be determined explicitly. For example, for p = 2 it is known that for the area of the isosceles right-angled triangle with the short sides of the length 1, the Poincaré constant C = 1 / π (and thus smaller than d / π of the diameter ).