A weak-strong convergence property and symmetry of minimizers of constrained variational problems in ${\mathbb{R}}^N$

We prove a weak-strong convergence result for functionals of the form ${\int }_{{\mathbb{R}}^N}j(x,u,Du)dx$ on $W^{1,p}$, along equiintegrable sequences. We will then use it to study cases of equality in the extended Polya-Szeg"o inequality and discuss applications of such a result to prove the symmetry of minimizers of a class of variational problems including nonlocal terms under multiple constraints.