[Authors] Bankevich, S. V.; Nazarov, A. I. [Title] On monotonicity of some functionals under rearrangements [AMS Subj-class] 49K05 Free problems in one independent variable 49N99 Calculus of variations and optimal control; optimization [Abstract] We consider the Polya--Szeg\"o type weighted inequality. We prove this inequality for monotone rearrangement and for Steiner's symmetrization. [Keywords] rearrangements, the Polya--Szeg\"o inequality, Lavrentiev's phenomenon [Comments] LaTeX, Russian, 23 pp. [Contact e-mail] al.il.nazarov@gmail.com