= Positivity criterion for a scalar gradient form
{title2=$\int_\Omega c|\nabla v|^2\geq0\ (v\in C_0^\infty)\ \Longleftrightarrow\ c\geq0\text{ a.e.}$}
Let $c$ be a bounded real measurable coefficient on an open set. Nonnegativity of $c$ clearly makes its gradient form nonnegative. Conversely, if $c<0$ on a set of positive measure, choose an interior <Lebesgue point> $x_0$ where $c(x_0)<0$. For a nonconstant compactly supported smooth function $\psi$, set $v_\varepsilon(x)=\varepsilon\psi((x-x_0)/\varepsilon)$. Change variables and use the <Lebesgue differentiation theorem> to obtain $\varepsilon^{-n}\int c|\nabla v_\varepsilon|^2\to c(x_0)\int|\nabla\psi|^2<0$, a contradiction. Thus the linear operator represented by $a(u,v)=\int c\nabla u\cdot\nabla v$ is monotone exactly when $c\geq0$ almost everywhere.
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