Anisotropic Least Gradient Problem

In [81], motivated by the Conductivity Imaging Problem, the authors introduced the following generalization of the least gradient problem \(\begin{aligned} \min \left\{ \int _\Omega \phi (x, Du): \quad u \in BV(\Omega ), \quad u\vert _{\partial \Omega } = f \right\} , \end{aligned}\) where \(\phi \) is a metric integrand (see below) and \(f \in C(\partial \Omega )\) . It is called the anisotropic least gradient problem. In this Chapter, we identify its associated Euler-Lagrange equation, which is an anisotropic counterpart of the 1-Laplace equation. In the first Section, we use a Gauss-Green formula to characterize the subdifferential of the relaxed energy functional and define the concept of weak solutions to the Dirichlet problem for the anisotropic 1-Laplacian. In the second Section, we introduce the notion of functions of \(\phi \) -least gradient and present some of their useful properties, including anisotropic counterparts of Miranda’s theorem and the Bombieri-De Giorgi-Giusti theorem proved in Chap. 1.