= Poincare inequality with a positive-measure zero set
{c}
{title2=$\|u\|_{L^2(B_R)}^2\leq C(n,\theta)R^2\|Du\|_{L^2(B_R)}^2$}
Suppose $u\in W^{1,2}(B_R)$ vanishes on a measurable subset of measure at least $\theta|B_R|$, with $\theta>0$. Writing $m$ for its mean, the zero set gives $\theta|B_R|m^2\leq\int_{B_R}|u-m|^2$. Thus $\int u^2\leq(1+\theta^{-1})\int|u-m|^2$, and the <Poincare-Wirtinger inequality> supplies the displayed estimate. An elementary proof of that inequality on a ball bounds differences along line segments and splits the segment parameter at $1/2$, giving the valid explicit constant $C(n,\theta)=2^{n+1}(1+\theta^{-1})$. The change of scale from the unit ball multiplies the constant by $R^2$.
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