An extension of the Hardy-Littlewood inequality for rearrangements is established. It is used for giving several conditions of existence of a minimum for nonweakly-lower-semicontinuous functionals of the form $J(v) \! = \! \int_0^1f(x,v(x),v'(x)) dx$ with constraints on $v$ and $v'$.
"Extended Hardy-Littlewood inequalities and applications to the calculus of variations." Adv. Differential Equations 5 (7-9) 1091 - 1138, 2000. https://doi.org/10.57262/ade/1356651295