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 precompact 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),

\| 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.

[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:

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},

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 for p = 1 is d/2. (See Acosta and Durán, 2004.)

[edit] References

  • Acosta, Gabriel and 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. 
  • 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:10.1137/S003614100343768X. ISSN 0036-1410.  MR2178227
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