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

2025/05/16 by Ficola, Eleonora, Schmidt, Thomas
#26B30 #35J20 #35R06 #49J45 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.11279

Abstract

We develop a semicontinuity-based existence theory in 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.

Citations

• Measures in the dual of BV: perimeter bounds and relations with divergence-measure fields
• Lower semicontinuity and existence results for anisotropic TV functionals with signed measure data
• The prescribed mean curvature measure equation in non-parametric form
• On a Neumann problem for variational functionals of linear growth
• Strict and pointwise convergence of BV functions in metric spaces
• Characterizations of signed measures in the dual of BV and related\n isometric isomorphisms
• The Mean Curvature Measure
• Sublinear functions of measures and variational integrals

Related

• Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints
• Lower semicontinuity and existence results for anisotropic TV functionals with signed measure data
• Relaxation and integral representation for functionals of linear growth\n on metric measure spaces
• Euler-Lagrange equations for variable-growth total variation
• Existence of W^1,1 solutions to a class of variational problems with linear growth on convex domains