We prove a strong continuity result for the relative rearrangement map. This new result and its corollary are used for the resolution of equations of the form $-\Delta u = F(u)$ via a Brouwer fixed point theorem. The nonlocal nonlinearity $F$ might depend on the monotone rearrangement $u_*$, its derivative $u'_*$ and the relative rearrangement of $u$ with respect to the data.
"Galerkin approximation, strong continuity of the relative rearrangement map and application to plasma physics equations." Differential Integral Equations 12 (1) 67 - 81, 1999. https://doi.org/10.57262/die/1367266994