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 the Friedrichs' inequality.
Statement of the inequality
The classical Poincaré inequality
Assume that 1 ≤ p ≤ ∞ and that Ω is bounded open subset of n-dimensional Euclidean space Rn having Lipschitz boundary (i.e., Ω is an open, bounded 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.
Generalizations
There exist 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 d/2 for p = 1, d/π for p = 2; (Acosta and Durán 2004) . 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 ( < d/π where ). (See, for instance,Kikuchi & Liu (2007) .)
References
- Acosta, Gabriel; Durán, Ricardo G. (2004), "An optimal Poincaré inequality in L1 for convex domains", Proc. Amer. Math. Soc., 132 (1): 195–202 (electronic), doi:10.1090/S0002-9939-03-07004-7
- Lawrence C., Evans (1998), Partial differential equations, Providence, RI: American Mathematical Society, ISBN 0-8218-0772-2
- Garroni, Adriana; Müller, Stefan (2005), "Γ-limit of a phase-field model of dislocations", SIAM J. Math. Anal., 36 (6): 1943–1964 (electronic), doi:10.1137/S003614100343768X, ISSN 0036-1410 MR2178227