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

Let p, so that 1 ≤ p < ∞ and Ω a subset with at least one bound. Then there exists a constant C, depending only on Ω and p, so that, for every function u of the Sobolev space W₀^1,p(Ω) of zero-trace functions,

\|u\|_{L^{p}(\Omega )}\leq C\|\nabla u\|_{L^{p}(\Omega )}.

Poincaré–Wirtinger inequality

Assume that 1 ≤ p ≤ ∞ and that Ω is a bounded connected open subset of the n-dimensional Euclidean space Rⁿ with a Lipschitz boundary (i.e., Ω is a 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. When Ω is a ball, the above inequality is called a (p,p)-Poincaré inequality; for more general domains Ω, the above is more familiarly known as a Sobolev inequality.

Generalizations

In the context of metric measure spaces (for example, sub-Riemannian manifolds), such spaces support a (q,p)-Poincare inequality for some $1\leq q,p<\infty$ if there are constants C and $\lambda \geq 1$ so that for each ball B in the space,

\mu (B)^{-{\frac {1}{q}}}\left\|u-u_{B}\right\|_{L^{q}(B)}\leq C\operatorname {rad} (B)\mu (B)^{-{\frac {1}{p}}}\|\nabla u\|_{L^{p}(\lambda B)}.

In the context of metric measure spaces, $|\nabla u|$ is the minimal p-weak upper gradient of u in the sense of Heinonen and Koskela [J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1–61]

There exist other 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|\left|{\hat {u}}(k)\right|^{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}{\operatorname {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 at most d/2 for p = 1, $d/\pi$ for p = 2 (Acosta & Durán 2004; Payne & Weinberger 1960), and this is the best possible estimate on the Poincaré constant in terms of the diameter alone. For smooth functions, this can be understood as an application of the isoperimetric inequality to the function's level sets. 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 $d={\sqrt {2}}$ ). (See, for instance,Kikuchi & Liu (2007).)

Furthermore, for a smooth, bounded domain $\Omega$ , since the Rayleigh quotient for the Laplace operator in the space $W_{0}^{{1,2}}(\Omega )$ is minimized by the eigenfunction corresponding to the minimal eigenvalue λ₁ of the (negative) Laplacian, it is a simple consequence that, for any $u\in W_{0}^{{1,2}}(\Omega )$ ,

\|u\|_{L^{2}}^{2}\leq \lambda _{1}^{-1}\left\|\nabla u\right\|_{L^{2}}^{2}

and furthermore, that the constant λ₁ is optimal.

References

Acosta, Gabriel; Durán, Ricardo G. (2004), "An optimal Poincaré inequality in L¹ for convex domains", Proc. Amer. Math. Soc., 132 (1): 195&ndash, 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
Kikuchi, Fumio; Liu, Xuefeng (2007), "Estimation of interpolation error constants for the P0 and P1 triangular finite elements", Comput. Methods. Appl. Mech. Engrg., 196 (37–40): 3750&ndash, 3758, doi:10.1016/j.cma.2006.10.029 MR 2340000
Garroni, Adriana; Müller, Stefan (2005), "Γ-limit of a phase-field model of dislocations", SIAM J. Math. Anal., 36 (6): 1943&ndash, 1964 (electronic), doi:10.1137/S003614100343768X MR 2178227
Leoni, Giovanni (2009), A First Course in Sobolev Spaces, Graduate Studies in Mathematics, American Mathematical Society, pp. xvi+607 ISBN 978-0-8218-4768-8, MR 2527916, Zbl 1180.46001, MAA
Payne, L. E.; Weinberger, H. F. (1960), "An optimal Poincaré inequality for convex domains", Archive for Rational Mechanics and Analysis: 286–292, ISSN 0003-9527
Heinonen, J.; Koskela, P. (1998), "Quasiconformal maps in metric spaces with controlled geometry", Acta Mathematica: 1–61, doi:10.1007/BF02392747, ISSN 1871-2509

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