# 庞加莱不等式

## 叙述

### 经典形式

p是一个大于等于1实数n是一个正整数。$\Omega$n欧几里得空间$\mathbb{R}^n$上的一个有界子集，并且其边界是满足利普希兹条件的区域（也就是说它的边界是一个利普希茨连续函数的图像）。在这种情况下，存在一个只与$\Omega$p有关的常数C，使得对索伯列夫空间$\mathbb{W}^{1,p}(\Omega)$ 中所有的函数u，都有：

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

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

### 推广

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

$\int_{\mathbf{T}^{2}} | u(x) |^{2} \, \mathrm{d} x \leq C \left( 1 + \frac1{\mathrm{cap} (E \times \{ 0 \})} \right) [ u ]_{H^{1/2} (\mathbf{T}^{2})}^{2},$

## 参考来源

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2. ^ Acosta, Gabriel; Durán, Ricardo G., An optimal Poincaré inequality in L1 for convex domains, Proc. Amer. Math. Soc., 2004, 132 (1): 195–202 (electronic), doi:10.1090/S0002-9939-03-07004-7
3. ^ M, Bebendorf, A Note on the Poincar´e Inequality for Convex Domains, Journal for Analysis and its Applications, 2003, 22 (4): 751–756
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