Schwarz Symmetrization Can Increase a Nonlocal Thereshold Energy

Abstract

For $N\geq 1$, $p\geq 1$, and $\delta>0$, consider the nonlocal threshold functional \[ I_{\delta,p}(u)=\iint_{\{|u(x)-u(y)|>\delta\}} \frac{\delta^p}{|x-y|^{N+p}}\,\dd x\,\dd y. \] Nguyen and Squassina \cite{NguyenSquassina} asked whether $I_{\delta,p}$ decreases under Schwarz rearrangement. We give a negative answer in every dimension. For each $N\geq1$ and $\delta>0$, one and the same explicit counterexample works for every $p\geq1$: it is nonnegative, bounded, compactly supported, and takes only five values. Its four nontrivial superlevel sets are nested balls whose centers alternate between two points. Once the active threshold interactions are isolated, the energy difference reduces to comparing the interaction of a unit ball with a concentric annulus and with an eccentric shell. The sign is strict because the potential generated by the unit ball decreases with the radius. We also compute the energy gap in closed form in dimension one. This settles Open Problem~2.2 of Nguyen and Squassina.