Existence of optimizers in a Sobolev inequality for vector fields

TL;DR

This paper proves the existence of minimizers for a Sobolev inequality involving vector fields by demonstrating the relative compactness of minimizing sequences, utilizing a specialized compactness theorem for nonlinear constraints.

Contribution

It establishes the existence of optimizers in a Sobolev inequality for vector fields, extending compactness results to nonlinear constrained sequences.

Findings

Minimizing sequences are relatively compact up to symmetries.

A version of the Rellich--Kondrachov theorem is adapted for nonlinear constraints.

Existence of a minimizer for the Sobolev inequality is confirmed.

Abstract

We consider the minimization problem corresponding to a Sobolev inequality for vector fields and show that minimizing sequences are relatively compact up to the symmetries of the problem. In particular, there is a minimizer. An ingredient in our proof is a version of the Rellich--Kondrachov compactness theorem for sequences satisfying a nonlinear constraint.