Mean-zero Poincare inequality on a bounded connected $C^1$ domain

Let $R={\prod }_{i=1}^3{I}_i$ be a box of positive side lengths and $u\in W^{1,2}(R^{\circ })$. For almost every pair $x,y\in R$, put $p_0=x$ and $p_i=(y_1,\dots ,y_i,x_{i+1},\dots ,x_3)$ for $1\le i\le 3$, so $p_3=y$. The telescoping sum gives $\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^2\le 3{\sum }_{i=1}^3\mathrm{\mid }u(p_{i-1})-u(p_i){\mathrm{\mid }}^2$. Along the coordinate segment from $p_{i-1}$ to $p_i$, ACL and Cauchy--Schwarz [L2] bound its squared difference by $\mathrm{\mid }{I}_i\mathrm{\mid }{\int }_{I_i}\mathrm{\mid }{\mathrm{\partial }}_{i}u(y_1,\dots ,y_{i-1},s,x_{i+1},\dots ,x_3){\mathrm{\mid }}^{2}ds$. Fubini makes all these statements valid for almost every pair; integrating in $x,y$ gives ${\int }_R{\int }_R\mathrm{\mid }u(p_{i-1})-u(p_i){\mathrm{\mid }}^{2}dxdy\le \mathrm{\mid }R\mathrm{\mid }\mathrm{\mid }{I}_i{\mathrm{\mid }}^2{\int }_R\mathrm{\mid }{\mathrm{\partial }}_{i}u{\mathrm{\mid }}^2$. Thus ${\int }_R{\int }_R\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^{2}dxdy\le C_R\mathrm{\mid }R\mathrm{\mid }{\int }_R\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2$ with $C_R=3{\max }_i\mathrm{\mid }{I}_i{\mathrm{\mid }}^2$.

The identity $u(x)-u_R=\mathrm{\mid }R{\mathrm{\mid }}^{-1}{\int }_R(u(x)-u(y))dy$ and averaged Cauchy--Schwarz [L2], integrated in $x$, give ${\int }_R\mathrm{\mid }u-u_R{\mathrm{\mid }}^2\le \mathrm{\mid }R{\mathrm{\mid }}^{-1}{\int }_R{\int }_R\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^{2}dxdy\le C_R{\int }_R\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2$. A nonempty half-box obtained by truncating the third interval at zero is itself a product of three intervals, so the same coordinate-line proof applies to it; no ACL assertion along arbitrary slanted segments is used.

Shrink each graph chart to the inverse image of a box compactly contained in its ambient chart range; its intersection $U$ with $\mathrm{\Omega }$ maps to a half-box $H$. For $v$ smooth near $\bar{\mathrm{\Omega }}$, $w=v\circ {\mathrm{\Phi }}^{-1}$ is $C^1$ on $\bar{H}$, which suffices for step 2.1. The mean $v_U$ minimizes ${\int }_U\mathrm{\mid }v-a{\mathrm{\mid }}^2$ over constants $a$: expanding about $v_U$ gives ${\int }_U\mathrm{\mid }v-a{\mathrm{\mid }}^2={\int }_U\mathrm{\mid }v-v_U{\mathrm{\mid }}^2+\mathrm{\mid }U\mathrm{\mid }\mathrm{\mid }a-v_U{\mathrm{\mid }}^2$. Choose $a=w_H$. The bounded Jacobian and inverse-chart derivatives [L3] then give ${\int }_U\mathrm{\mid }v-v_U{\mathrm{\mid }}^2\le {\int }_U\mathrm{\mid }v-w_H{\mathrm{\mid }}^2\le C{\int }_H\mathrm{\mid }w-w_H{\mathrm{\mid }}^2\le C_U{\int }_U\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2$. Interior boxes obey step 2.1 directly.

The ambient chart boxes of step 3.1 and interior boxes cover the compact closure, so a finite subcover gives $\mathrm{\Omega }={\bigcup }_{j=1}^N{U}_j$ with nonempty open patches. Every nonempty overlap is open in ${\mathbb{R}}^3$, contains a box, and has positive measure. The graph joining nonempty overlaps is connected: otherwise its component unions would partition the connected $\mathrm{\Omega }$ into disjoint nonempty relatively open sets [L4]. Choose ambient smooth approximants $v_m\to v$ in $H^1(\mathrm{\Omega })$ using density [L3] and Countable Choice [L5]. Restrictions and gradients converge in $L^2$, and $\mathrm{\mid }(v_m{)}_{U_j}-v_{U_j}\mathrm{\mid }\le \mathrm{\mid }{U}_j{\mathrm{\mid }}^{-1\mathrm{/}2}\mathrm{\parallel }{v}_m-v{\mathrm{\parallel }}_{L^2(U_j)}$ by [L2]. Hence step 3.1 passes to the limit, giving ${\int }_{U_j}\mathrm{\mid }v-v_{U_j}{\mathrm{\mid }}^2\le C_j{\int }_{U_j}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2$.

For an edge $(j,k)$, integrating $\mathrm{\mid }{v}_{U_j}-v_{U_k}{\mathrm{\mid }}^2\le 2\mathrm{\mid }{v}_{U_j}-v(x){\mathrm{\mid }}^2+2\mathrm{\mid }v(x)-v_{U_k}{\mathrm{\mid }}^2$ over its positive-measure overlap and using step 4.1 bounds the squared mean difference by a cover-dependent constant times ${\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2$. A path of at most $N-1$ edges from $1$ to $j$ exists by connectedness; the squared sum of its differences is at most the path length times the sum of their squares. Thus $\mathrm{\mid }{v}_{U_j}-v_{U_1}{\mathrm{\mid }}^2\le C^{\mathrm{\prime }\mathrm{\prime }}{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2$ for every $j$, also for $N=1$ when the difference is zero.

The constant-minimization identity of step 3.1 on $\mathrm{\Omega }$ gives ${\int }_{\mathrm{\Omega }}\mathrm{\mid }v-v_{\mathrm{\Omega }}{\mathrm{\mid }}^2\le {\int }_{\mathrm{\Omega }}\mathrm{\mid }v-v_{U_1}{\mathrm{\mid }}^2\le {\sum }_j{\int }_{U_j}\mathrm{\mid }v-v_{U_1}{\mathrm{\mid }}^2\le 2{\sum }_j{\int }_{U_j}\mathrm{\mid }v-v_{U_j}{\mathrm{\mid }}^2+2{\sum }_j\mathrm{\mid }{U}_j\mathrm{\mid }\mathrm{\mid }{v}_{U_j}-v_{U_1}{\mathrm{\mid }}^2$. Step 4.1 bounds the first sum by $2{\sum }_j{C}_j{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2$, and step 5.1 bounds the second by a cover-dependent multiple of the same integral. Taking square roots gives the stated estimate with a constant depending only on the fixed domain and cover. The choice uses are those recorded in [L5].