Qinyang Li

Schwarz Symmetrization Can Increase a Nonlocal Thereshold Energy

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...