Fix $μ>0$ and let $U$ be an open, bounded, connected set, $U⊂R^n$. We use the usual notation of $S$ to denote the Lebesgue measure of a set $S⊂R^n$. If $|\left\{x∈U | u(x)=0\right\}|≥μ>0$ and $u∈H_0^1 (U)$, then show there exists a constant $C$, which depends on $n$, $μ$, and $U$, such that
$$∫_U u^2 dx ≤C∫_U|Du|^2 dx$$
So after doing some digging on Poincare's inequality I am still not sure how I would provide a proof for this.