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

TL;DR

This paper establishes a weak-strong convergence result for certain functionals in variational calculus and applies it to demonstrate symmetry properties of minimizers, including cases with nonlocal terms and multiple constraints.

Contribution

It introduces a new weak-strong convergence theorem for functionals on Sobolev spaces and applies it to prove symmetry of minimizers in constrained variational problems.

Findings

Proved a weak-strong convergence result for integral functionals.

Demonstrated symmetry of minimizers in variational problems with nonlocal terms.

Extended Polya-Szeg"o inequality cases analyzed.

Abstract

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.