Lower semicontinuity for an integral functional in BV

TL;DR

This paper establishes a lower semicontinuity result for a linear growth integral functional defined on BV functions with respect to a measure that is absolutely continuous with respect to Lebesgue measure.

Contribution

It extends lower semicontinuity results to a broader class of measures for BV functionals with linear growth, under minimal assumptions.

Findings

Proves lower semicontinuity for BV functionals with linear growth.

Results apply to measures absolutely continuous with respect to Lebesgue measure.

Provides a framework for analyzing variational problems with general measures.

Abstract

We prove a lower semicontinuity result for a functional of linear growth initially defined by \[ \int_{\Omega}F\left(\frac{dDu}{d\mu}\right)\,d\mu \] for $u\in BV(\Omega;\mathbb{R}^N)$ with $Du\ll \mu$. The positive Radon measure $\mu$ is only assumed to satisfy $\mathcal L^n\ll \mu$.