Let $U\subset\mathbb{R}^n$ be bounded and open. Let $L$ be a divergence form [elliptic operator](/page/Elliptic%20Operator) with $b_i=0$ for $1\le i\le n$, with $c\ge 0$ $\mathcal{L}^n$-a.e. in $U$, and with leading coefficients satisfying the ellipticity bound with constant $\theta>0$. Let $B:H_0^1(U)\times H_0^1(U)\to\mathbb R$ denote the associated bilinear form,

\begin{align*} B[u,v] &=\int_U \sum_{i,j=1}^n a_{ij}\partial_{x_j}u\,\partial_{x_i}v+cuv\,d\mathcal L^n . \end{align*}

Then there exists $\alpha>0$, depending on $U$ and $\theta$, such that

\begin{align*} B[u,u]\ge \alpha\|u\|_{H_0^1(U)}^2 \end{align*}

for every $u\in H_0^1(U)$.