Existence of parabolic minimizers to the total variation flow on metric measure spaces

On the Heat flow on metric measure spaces: existence, uniqueness and stability

We prove existence and uniqueness of the gradient flow of the Entropy functional under the only assumption that the functional is λ-geodesically convex for some λ ∈ R. Also, we prove a general stability result for gradient flows of geodesically convex functionals which Γ−converge to some limit functional. The stability result applies directly to the case of the Entropy functionals on compact sp...

Inhomogeneous parabolic equations on unbounded metric measure spaces

In recent years, the study of partial differential equations on self-similar fractals has attracted increasing interest (see, for example, [7–9, 13, 14]). We investigate a class of nonlinear diffusions with source terms on general metric measure spaces. Diffusion is of fundamental importance in many areas of physics, chemistry and biology. Applications of diffusion include sintering, i.e. makin...

iteration for ( quasi ) minimizers on metric spaces

We study regularity properties of quasiminimizers of the p-Dirichlet integral on metric measure spaces. We adapt the Moser iteration technique to this setting and show that it can be applied without an underlying differential equation. However, we have been able to run the Moser iteration fully only for minimizers. We prove Caccioppoli inequalities and local boundedness properties for quasisuba...

Equivalent definitions of BV space and of total variation on metric measure spaces

Sobolev and Bounded Variation Functions on Metric Measure Spaces

Contents Chapter 1. Introduction 1 1. History 1 2. Motivations 3 3. Examples of metric measure spaces 4 Chapter 2. H-Sobolev space and first tools of differential calculus 9 1. Relaxed slope and Cheeger energy 9 2. Elements of differential calculus 11 3. Reminders of convex analysis 14 4. Laplacian and integration by parts formula 15 5. Heat flow in (X, d, m) 16 Chapter 3. The Lagrangian (Beppo...