Seminari del Dipartimento di Matematica

Seminario del 2024

Eleonora Ficola

Minimizing anisotropic total variation functionals depending on measures

analisi matematica

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.