# $\Gamma$-convergence for functionals depending on vector fields. I. Integral representation and compactness

## TL;DR

This paper investigates the $ ext{Gamma}$-convergence of functionals depending on locally Lipschitz vector fields, establishing integral representations and compactness results crucial for understanding variational problems involving such vector fields.

## Contribution

It provides the first integral representation and $ ext{Gamma}$-compactness results for functionals depending on locally Lipschitz vector fields, extending variational analysis in this context.

## Key findings

- Established integral representation for local functionals depending on vector fields.
- Proved $ ext{Gamma}$-compactness for a class of integral functionals involving vector fields.
- Extended the theoretical framework for variational problems with vector field dependence.

## Abstract

Given a family of locally Lipschitz vector fields $X(x)=(X_1(x),\dots,X_m(x))$ on $\mathbb{R}^n$, $m\leq n$, we study functionals depending on $X$. We prove an integral representation for local functionals with respect to $X$ and a result of $\Gamma$-compactness for a class of integral functionals depending on $X$.