This chapter is dedicated to the study of the total variation flow on the whole space. In order to use the Anzellotti pairings and the Gauss-Green formula introduced in Chapter 4, we suppose that the metric space is complete, separable, equipped with a doubling measure, and that it supports a weak Poincaré-type inequality. The total variation flow in this setting is understood as the gradient flow of the 1-Cheeger energy, which (as shown by Ambrosio and Di Marino) is convex and lower semicontinuous with respect to L2 convergence; then, the Brezis–Komura theorem guarantees the existence of a unique strong solution. We now provide a characterisation of the subdifferential of the 1-Cheeger energy, introduce the notion of weak solution to the total variation flow based on this characterisation, and prove their existence and uniqueness. We also comment on the asymptotic behaviour of weak solutions and introduce a notion of entropy solutions for initial data in L1 under the assumption that the measure of the space is finite.

In this chapter, we provide an overview of several related problems in which we can apply the techniques developed in Chapters 2–6. We first study the least gradient problem, that is, the problem of minimisation of total variation with respect to a given Dirichlet boundary condition. We discuss several possible formulations of this problem in a metric measure setting, and in regular bounded open subsets of a metric measure space, we describe the relation between the least gradient problem and the Dirichlet problem for the 1-Laplacian operator. Section 7.2 deals with the Cheeger problem and the Cheeger cut problem in the general framework of metric measure spaces. Our first goal is to study the problem of characterising the Cheeger constant of a bounded domain and the identification of the Cheeger sets. Furthermore, we prove a version of the Max-Flow Min-Cut Theorem. Then, we consider the Cheeger cut problem of partitioning the space into two parts so that the Cheeger constant of the whole space is achieved, and as a consequence, we obtain a Cheeger inequality along the lines of the classical one for Riemannian manifolds obtained by Cheeger in 1970.

In this chapter, we study the p-Laplacian evolution equation for p > 1, that is, the gradient flow of the p-Cheeger energy. Our goal is to use the Hilbertian theory of gradient flows to provide a notion of weak solution to the gradient flow associated to the p-Cheeger energy and prove their existence and uniqueness for initial data in L2. We assume that the metric space is complete and separable and that the associated measure is finite on bounded sets (we keep this assumption throughout the whole chapter). Then, Ambrosio, Gigli, and Savaré proved that the p-Cheeger energy is convex and lower semicontinuous; since it’s the domain of definition is dense in L2 as it contains Lipschitz functions with bounded support, by the Brezis–Komura theorem, for every initial data in L2, there exists a unique strong solution to the abstract Cauchy problem. Our goal is to introduce a notion of weak solutions, improving upon the above definition; we will express the subdifferential of the p-Cheeger energy in terms of the differential structure due to Gigli presented in Chapter 1.

This chapter is devoted to presenting some basic results concerning the differential structure on an arbitrary metric space. By a metric measure space, we mean a complete and separable metric space equipped with a non-zero non-negative Borel measure which is finite on bounded sets. In the literature, there are several possible definitions of Sobolev spaces in this setting, most prominently via p-upper gradients, p-relaxed slopes, and via test plans. Under minimal assumptions, all these definitions agree; we present these equivalent approaches in Appendix B and now introduce the most suitable definition for our purposes: the Newtonian space. We also present the definition of functions of bounded variation, their main properties, and discuss how to define the boundary measure of a set of finite perimeter and the relationships between them. In the second half of this chapter, we present the theory of Lp-normed modules and the first-order differential structure on metric measure spaces introduced by Gigli. We also recall the definition of divergence in the metric setting and discuss how to adapt these notions to the case of a sufficiently regular open subset of a metric measure space.

Over the last 30 years, a number of possible definitions of Sobolev and BV spaces in the metric setting have been proposed by various authors, and a priori it is not known whether they are equivalent. In this monograph, we decided to work with the definition of Newtonian spaces proposed by Shanmugalingam. In this appendix, we present several other approaches present in the literature and comment on their relationship and dependence on the exponent. The first general result concerning the equivalence of the most common definitions of Sobolev spaces was proved by Ambrosio, Gigli, and Savaré in the case p > 1; similar results were subsequently shown for the BV spaces and for the Sobolev space with exponent equal to one. To give a complete historical overview, we present here several variants of definitions and several equivalence results. We also discuss the dependence of the minimal p-weak upper gradient on the exponent. Throughout this appendix, as in the whole monograph, we assume that the metric space is complete, separable, and equipped with a non-negative Borel measure which is finite on bounded subsets.

Our next aim is to extend the results of Chapter 2 and introduce a notion of weak solution to gradient flows in metric measure spaces in a fairly general setting. Our main assumption is that the functional only depends on the differential of a function. In particular, this setting covers the case when the functional only depends on the function through its minimal p-weak upper gradient. In this entire chapter, we assume that p > 1 and that we work with a convex and lower semicontinuous functional defined on L2, which is given by a composition of the differential and a non-negative, continuous, convex, and coercive functional defined on the cotangent space. We first present the general framework under the minimal structural assumptions described above. Then, we apply the newly developed techniques to study a specific functional with inhomogeneous growth, which is the sum of two Cheeger energies for different exponents.

We now study the total variation flow on bounded domains in metric measure spaces. In Section 6.1, we consider the Neumann problem; using the techniques developed in Chapters 4 and 5, we give a definition of weak solution to the Neumann problem for initial data in L2 based on the Gigli differential structure adapted to a bounded domain and prove their existence and uniqueness. We also introduce the notion of entropy solution for initial data in L1. In Section 6.2, we consider the Dirichlet problem for initial data in L2 and boundary data in L1. We prove lower semicontinuity of the associated functional, give a definition of weak solution, and prove their existence and uniqueness.