The Anzellotti–Gauss–Green formula and least gradient functions in metric measure spaces

Abstract

In the framework of the first-order differential structure introduced by Gigli, we obtain a Gauss–Green formula on regular bounded open sets in doubling metric measure spaces supporting a weak Poincaré inequality, valid for BV functions and vector fields with integrable divergence. Then, we study least gradient functions in metric measure spaces and provide an Euler–Lagrange-type formulation of the least gradient problem, using this formula as the main tool.