Superminimizers and a weak Cartan property for \(p = 1\) in metric spaces (Q2177960)

The paper deals with `1-minimizers', i.e., functions of least gradients in metric spaces endowed with a doubling measure and for which a Poincaré inequality holds. By means of suitable obstacle problems, De Giorgi-type and weak Harnack inequalities for their solutions and \(1\)-subminimizers, semicontinuity properties for upper and lower approximate limits of \(1\)-superminimizers are provided. The obtained results also generalize and recover previous results dealing with the continuity of the precise representatives. Properties of the fine topology for \(p = 1\) are obtained. Moreover, a weak Cartan property for superminimizers is proven and used to show that any topology for which the upper representative of every \(1\)-superminimizer is upper semicontinuous in open sets is stronger than the \(1\)-fine topology. The main contributions of the article are summarized at the end of the paper by means of a useful schematic comparison between the properties of Newton-Sobolev, \(p\)-super-harmonic functions, and BV, \(1\)-superharmonic functions.