Abstract:

In my habilitation thesis, I study functionals that involve the so-called anisotropictotal variation. The results can be divided into two groups. In the first part of thethesis, I study the gradient flows of such functionals in metric measure spaces usinga semigroup approach. In particular, I investigate the impact of the underlyinggeometry on the behavior of solutions. In the second part of the thesis, I focus onthe problem of minimizing the anisotropic total variation under Dirichlet conditionsin dimension two.

In this lecture, I will present results from the second part of my thesis. Due to thelinear growth of the functionals we consider, the existence of solutions is notguaranteed as the boundary condition may not be attained pointwise. In fact, theexistence, the uniqueness, and the regularity of solutions depend on the specificanisotropy, the geometry of the domain, and the boundary data. Expanding on theequivalence between the least gradient problem and the optimal transportproblem, we study the regularity and the stability of solutions in the anisotropiccase. We also discuss the dependence of the set of admissible boundary data onthe anisotropy, showing that the trace spaces associated with any two differentsmooth anisotropies are also distinct.