A differentiable manifold, defined with the help of collections of charts, comes with basic notions of calculus prior to the introduction of the metric. The most important is how one may define vectors as directional derivatives, 1-forms via their dual pair, and build up general tensors from these two. Partial derivatives for functions extend to the Lie derivative, while a special subclass of tensors known as differential forms admits another type of derivative known as the exterior derivative. We develop calculus based on these most basic differential structures, ending with the Stokes theorem. Once a differentiable manifold is given, one can turn to a further structure of a metric and its accompanying Levi-Civita connection, leading to the Riemann curvature tensor. Numerous additional structures made possible by the covariantly constant metric are introduced, such as raising and lowering of indices, Killing vector fields, the volume form, the Hodge star map, geodesics, and geodesic normal coordinates.

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.

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.

This chapter is primarily a preparation for the study of the total variation flow. This is due to the fact that in order for the 1-Cheeger energy to be lower semicontinuous, it needs to be defined on the space of functions of bounded variation and not on the Sobolev space with exponent equal to one. Thus, we need to extend parts of the linear differential structure to the BV case, and to this end, we will require that we can approximate BV functions with Lipschitz functions in a suitable way. Our strategy will be to present a metric version of the Anzellotti pairings between a vector field with integrable divergence and a BV function due to Anzellotti and prove a generalised integration by parts formula. To be precise, we will prove the existence of two such pairings, separately for the case of the whole space and for a bounded domain with a sufficiently regular boundary. They are defined slightly differently, in particular, in the second case, the Gauss-Green formula takes into account the boundary effects.

A differentiable manifold, defined with the help of collections of charts, comes with basic notions of calculus before the introduction of the metric. We start with the definitions of vectors as directional derivatives and 1-forms via the natural dual pairing and build up general tensors from these two. Partial derivatives for functions extend to the Lie derivative, while a special subclass of tensors known as differential forms admits the so-called exterior derivative. We develop calculus based on these most basic structures, ending with the Stokes theorem. This sets the stage for the Riemannian geometry, given in two alternate forms in Chapters 4 and 5.