Schwarz Symmetrization Can Increase a Nonlocal Thereshold Energy

Qinyang Li

For $N\ge 1$, $p\ge 1$, and $\delta >0$, consider the nonlocal threshold functional \[ I_{δ,p}(u)=\iint_{\{|u(x)-u(y)|>δ\}} \frac{δ^p}{|x-y|^{N+p}}\,\dd x\,\dd y. \] Nguyen and Squassi…