Lower semicontinuity and existence results for anisotropic total variation functionals with measure data

Abstract

Aim of the talk is to present an existence result to the anisotropic 1-Laplace problem
Div ∇ξφ(· , ∇u) = μ on Ω
with Dirichlet boundary datum u0 ∈ L1 (∂Ω) and μ a signed, Radon measure on Ω. Our approach consists in proving the existence of BV-minimizers for the corresponding integral functional Φu0. In doing so, we characterize the appropriate assumptions for the measure μ in order to obtain lower-semicontinuity of the anisotropic functional Φu0, and discuss a refined LSC in the related parametric case. We further prove that the definition of Φu0 is consistent with the original one set in the Sobolev space W1,1 u0 (Ω) and provide some illustrative examples. Finally, further research directions will be sketched to include a broader class of functionals with linear growth.

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