Generalized $\mathbf{W^{1,1}}$-Young measures and relaxation of problems with linear growth

TL;DR

This paper fully characterizes generalized Young measures generated by gradient sequences in $W^{1,1}$, extending previous work to include boundary concentrations, and applies these results to relax non-quasiconvex variational problems with linear growth.

Contribution

It extends the characterization of Young measures to include boundary concentrations and applies this to the relaxation of variational problems with linear growth.

Findings

Complete characterization of generalized Young measures in $W^{1,1}$.

Extension of previous analysis to include boundary concentrations.

Application to relaxation of non-quasiconvex variational problems.

Abstract

We completely characterize generalized Young measures generated by sequences of gradients of maps from $W^{1,1}(\Omega;\R^M)$ where $\Omega\subset\R^N$. This extends and completes previous analysis by Kristensen and Rindler where concentrations of the sequence of gradients at the boundary of $\Omega$ were excluded. We apply our results to relaxation of non-quasiconvex variational problems with linear growth at infinity. We also link our characterization to Sou\v{c}ek spaces \cite{soucek}, an extension of $W^{1,1}(\Omega;\R^M)$ where gradients are considered as measures on $\bar\Omega$.