Poincaré inequality

In analysis , the Poincaré inequality is an inequality from the theory of Sobolev spaces named after the French mathematician Henri Poincaré . The inequality makes it possible to derive bounds for a function from bounds of the derivatives and the geometry of the domain. Such bounds play an important role in the calculus of variations .

Formulation of the inequality

The classic Poincaré inequality

Let and be a bounded connected open subset of the - dimensional Euclidean space with Lipschitz boundary (i.e. is a Lipschitz domain ). Then there is a constant that only depends on and , so that for every function in Sobolev space the inequality ${\displaystyle 1\leq p<\infty }$ ${\displaystyle \Omega }$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \Omega }$ ${\displaystyle C}$ ${\displaystyle \Omega }$ ${\displaystyle p}$ ${\displaystyle u}$ ${\displaystyle W^{1,p}(\Omega )}$

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

holds, where

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

the mean of over is denotes the Lebesgue measure of the area . ${\displaystyle u}$ ${\displaystyle \Omega }$ ${\displaystyle |\Omega |}$ ${\displaystyle \Omega }$

With the help of the Hölder inequality one can show that the -Poincaré inequality follows from the -Poincaré inequality. General: If the Poincaré inequality applies to one for a region , then it also applies to all of them , possibly with a different constant . ${\displaystyle L^{2}}$ ${\displaystyle L^{1}}$ ${\displaystyle \Omega }$ ${\displaystyle p^{\prime }}$ ${\displaystyle p>p^{\prime }}$ ${\displaystyle C}$

One-dimensional example

Let f be a continuously differentiable function with a Fourier series

${\displaystyle f(x)=\sum _{n=-\infty }^{\infty }a_{n}e^{inx}}$,

then is using Parseval's equation

${\displaystyle \|f-f_{[0,2\pi ]}\|_{L^{2}[0,2\pi ]}=\sum _{n\not =0}|a_{n}|^{2}\leq \sum _{n}n^{2}|a_{n}|^{2}=\|f'\|_{L^{2}[0,2\pi ]}}$.

Manifolds

For Riemannian manifolds with nonnegative Ricci curvature (for example nonnegative sectional curvature ) the Poincaré inequality applies. There is one constant that only depends on the dimension n , so that applies to all : ${\displaystyle C_{n}}$ ${\displaystyle p\in M,r>0,u\in W_{loc}^{2,1}(M)}$

${\displaystyle \|u-u_{B(p,r)}\|_{L^{2}(B(p,r))}\leq C_{n}r\|\nabla u\|_{L^{2}(B(p,r))}}$

Metric spaces

In 2007 Bruce Kleiner proved a Poincaré inequality for the Cayley graphs of finitely generated groups:

${\displaystyle \int _{B_{R}}\|f-f_{R}\|^{2}\leq 8\|S\|^{2}R^{2}{\frac {\|B(2R)\|}{\|B(R)\|}}\int _{B_{3R}}\|\nabla f\|^{2},}$

where is a piecewise smooth function, its mean over the ball and the generating system defining the Cayley graph. Using this inequality, he gave a simplified proof of Gromow's theorem on groups of polynomial growth. ${\displaystyle f}$ ${\displaystyle f_{R}}$ ${\displaystyle B_{R}}$ ${\displaystyle S}$

For metric spaces with non-negative Ricci curvature in the sense of Lott-Villani-Sturm , the weak local -Poincaré inequality was proven by Rajala in 2012. ${\displaystyle L^{1}}$

Generalizations

There are generalizations of the Poincaré inequality for other Sobolev spaces, for example the following Poincaré inequality for Sobolev space , i. H. the space of the functions in the -space of the torus , whose Fourier transform the condition ${\displaystyle H^{\frac {1}{2}}(T^{2})}$ ${\displaystyle u}$ ${\displaystyle L^{2}}$ ${\displaystyle T^{2}}$ ${\displaystyle {\hat {u}}}$

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

met: There is a constant , so that for each with the same 0 to an open set following inequality holds: ${\displaystyle C}$ ${\displaystyle u\in H^{\frac {1}{2}}(T^{2})}$ ${\displaystyle u}$ ${\displaystyle E\subset T^{2}}$

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

where the harmonic capacity of means as a subset of . ${\displaystyle \operatorname {cap} (E\times \left\{0\right\})}$ ${\displaystyle E\times \left\{0\right\}}$ ${\displaystyle \mathbb {R} ^{3}}$

The Poincaré constant

The optimal constant in the Poincaré inequality is called the Poincaré constant of the area . It is generally very difficult to determine the Poincaré constant, depending on and the geometry of the area . However, certain special cases can be treated. For example, for restricted, convex Lipschitz regions with a diameter , the Poincaré constant is at most if , and at most if, and that is the best possible estimate for the Poincaré constant that depends only on the diameter. For smooth functions this is obtained as an application of the isoperimetric inequality to the level sets of the function. In the one-dimensional, this is the Wirtinger inequality for functions. ${\displaystyle C}$ ${\displaystyle \Omega }$ ${\displaystyle p}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle d}$ ${\displaystyle d/2}$ ${\displaystyle p=1}$ ${\displaystyle d/\pi }$ ${\displaystyle p=2}$

There are special cases in which the constant can be determined explicitly. For example , it is known that for the area of ​​the isosceles right triangle with cathets of length 1 the Poincaré constant is (and thus smaller than for the diameter ). ${\displaystyle C}$ ${\displaystyle p=2}$ ${\displaystyle C=1/\pi }$ ${\displaystyle d/\pi }$ ${\displaystyle d={\sqrt {2}}}$

Individual evidence

1. ^ Peter Buser: A note on the isoperimetric constant . In: Ann. Sci. École Norm. Sup. , (4) 15, 1982, no. 2, pp. 213-230
2. ↑ Bruce Kleiner: A new proof of Gromov's theorem on groups of polynomial growth . In: J. Amer. Math. Soc. , 23, 2010, no. 3, pp. 815-829, arxiv : 0710.4593
3. ^ Tapio Rajala: Local Poincaré inequalities from stable curvature conditions on metric spaces . In: Calc. Var. Partial Differ. Equ. , 44, No. 3-4, 2012, pp. 477-494
4. ^ Adriana Garroni, Stefan Müller: -limit of a phase-field model of dislocations . In: SIAM J. Math. Anal. , 36, 2005, no. 6, pp. 1943-1964
5. ^ Gabriel Acosta, Ricardo Durán: An optimal Poincaré inequality in for convex domains ${\displaystyle L^{1}}$ . In: Proc. Amer. Math. Soc. , 132, 2004, no. 1, pp. 195-202
6. ^ LE Payne, Hans F. Weinberger: An optimal Poincaré inequality for convex domains . In: Arch. Rational Mech. Anal. , 5, 1960, pp. 286-292.
7. ^ Nick Alger: L1 Poincaré Inequality .
8. ↑ Fumio Kikuchi, Xuefeng Liu: Estimation of interpolation error constants for the P0 and P1 triangular finite elements . In: Comput. Methods Appl. Mech. Eng. , 196, 2007, no. 37-40, pp. 3750-3758.