Poincare inequality with a positive-measure zero set ($\Vert u{\Vert }_{L^2(B_R)}^2\le C(n,\theta )R^2\Vert Du{\Vert }_{L^2(B_R)}^2$)

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 $\theta |B_R|m^2\le {\int }_{B_R}|u-m{|}^2$. Thus $\int u^2\le (1+{\theta }^{-1})\int |u-m{|}^2$, 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 $R^2$.