Existence theory for linear-growth variational integrals with signed measure data

TL;DR

This paper establishes an existence theory for scalar linear-growth variational integrals with signed measure data in BV spaces, extending previous results and enabling variational analysis of related Euler-Lagrange equations.

Contribution

It develops a semicontinuity-based existence framework for variational integrals with signed measures, refining prior work on anisotropic total variations and area-type cases.

Findings

Proves existence of solutions in BV for a broad class of variational integrals.

Extends previous theories to include signed measure data.

Facilitates variational analysis of Euler-Lagrange equations with measure data.

Abstract

We develop a semicontinuity-based existence theory in $\mathrm{BV}$ for a general class of scalar linear-growth variational integrals with additional signed-measure terms. The results extend and refine previous considerations for anisotropic total variations and area-type cases, and they pave the way for a variational approach to the corresponding Euler-Lagrange equations, which involve the signed measure as right-hand-side datum.