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 W^1,p(Ω),

$\|u-u_{\Omega }\|_{L^{p}(\Omega )}\leq C\|\nabla u\|_{L^{p}(\Omega )},$

where

$u_{\Omega }={\frac {1}{|\Omega |}}\int _{\Omega }u(y)\,\mathrm {d} y$

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 H^1/2(T²), i.e. the space of functions u in the L² space of the unit torus T² with Fourier transform û satisfying

$[u]_{H^{1/2}(\mathbf {T} ^{2})}^{2}=\sum _{k\in \mathbf {Z} ^{2}}|k|{\big |}{\hat {u}}(k){\big |}^{2}<+\infty :$

there exists a constant C such that, for every u ∈ H^1/2(T²) with u identically zero on an open set E ⊆ T²,

$\int _{\mathbf {T} ^{2}}|u(x)|^{2}\,\mathrm {d} x\leq C\left(1+{\frac {1}{\mathrm {cap} (E\times \{0\})}}\right)[u]_{H^{1/2}(\mathbf {T} ^{2})}^{2},$

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 $\scriptstyle {d={\sqrt {2}}}$ ). (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

• Fumio, Kikuchi; Xuefeng, Liu (2007), "Estimation of interpolation error constants for the P0 and P1 triangular finite elements", Comput. Methods. Appl. Mech. Engrg., 196: 3750–3758, doi:10.1016/j.cma.2006.10.029, ISSN 0045-7825 MR2340000