Past exam of the methods of applied functional analysis course of the University of Oxford / 2010 / B / 6 / c / i / Solution

The operator is well-defined in every dimension: $|\cos w|\le 1$ givesFor fixed $w$, its monotonicity condition isThus $\cos w\ge 0$ almost everywhere is sufficient. It is also necessary, by the positivity criterion for a scalar gradient form. Indeed, if the bounded coefficient $c=\cos w$ is negative on a set of positive measure, choose a Lebesgue point $x_0$ with $c(x_0)<0$ and a nonconstant compactly supported smooth $\psi$. Set $z_{\epsilon }(x)=\epsilon \psi ((x-x_0)/\epsilon )$, supported inside $\Omega$. Thencontradicting monotonicity. Therefore $\boxed{A(w,\cdot )\text{ is monotone exactly when }\cos w\ge 0\text{ almost everywhere}}$. In particular it is not monotone for every $w$: the admissible constant $w\equiv \pi$ makes it the negative Dirichlet form.