Generalised Young Measures and characterisation of gradient Young Measures

TL;DR

This paper introduces a new representation called generalized Young Measures to describe the limits of integrals involving bounded sequences of functions, and characterizes those generated by gradients through integral inequalities.

Contribution

It develops a novel framework for representing accumulation points of integrals using generalized Young Measures and characterizes gradient-generated measures via inequalities.

Findings

New representation of accumulation points for functions of linear growth.

Characterization of gradient Young Measures through integral inequalities.

Provides a comprehensive framework for analyzing limits of bounded sequences.

Abstract

Given a function $f\in C(\mathbb{R}^d)$ of linear growth, we give a new way of representing accumulation points of \begin{equation} \int_\Omega f(v_i(z))d\mu(z), \end{equation} where $\mu\in \mathcal{M}^+(\Omega)$, and $(v_i)_{i\in \mathbb{N}}\subset L^1(\Omega,\mu)$ is norm bounded. We call such representations "generalised Young Measures". With the help of the new representations, we then characterise these limits when they are generated by gradients, i.e. when $v_i = Du_i$ for $u_i\in W^{1,1}(\Omega,\mathbb{R}^m)$, via a set of integral inequalities.