Poincaré inequality
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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.
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[edit] Statement of the inequality
[edit] 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.
[edit] 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
![[ u ]_{H^{1/2} (\mathbf{T}^{2})}^{2} = \sum_{k \in \mathbf{Z}^{2}} | k | \big| \hat{u} (k) \big|^{2} < + \infty:](http://upload.wikimedia.org/math/4/b/f/4bf5bd79e3ea78e00d33bfa1255c536f.png)
there exists a constant C such that, for every u ∈ H1/2(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 + \frac1{\mathrm{cap} (E \times \{ 0 \})} \right) [ u ]_{H^{1/2} (\mathbf{T}^{2})}^{2},](http://upload.wikimedia.org/math/5/f/c/5fcdf26b722555d66002d13744debfca.png)
where cap(E × {0}) denotes the harmonic capacity of E × {0} when thought of as a subset of R3.
[edit] 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).)
[edit] 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:
- Evans, Lawrence C. (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: MR2178227
