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

We develop a semicontinuity-based existence theory in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>BV</m:mi> </m:math> {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.