Minimizing anisotropic total variation functionals depending on measures

Aim of the talk is to present an existence result to the anisotropic 1-Laplace problem div [∇_ξ φ(·,∇u)] = μ on Ω with Dirichlet boundary datum u_0 in L^1(∂ Ω) and μ a signed, Radon measure on Ω. Our approach consists in proving the existence of BV-minimizers for the corresponding integral functional Φ_{u_0}. In doing so, we characterize the appropriate assumptions for the measure μ in order to obtain lower-semicontinuity of Φ_{u_0}, and discuss a refined LSC for the related parametric functional. Additionally, we prove the definition of Φ_{u_0} to be consistent with the original anisotropic problem in the Sobolev space W^{1,1}_{u_0}(Ω) and provide some examples. Finally, further research directions will be sketched to include a broader class of functionals with linear growth.