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

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.

💡 Research Summary

The paper addresses the variational problem of minimizing a functional that combines a first‑order linear‑growth integral with a signed Radon measure term, set in the space of functions of bounded variation (BV). The functional under consideration is

 F₍µ₎^{u₀}