Two classes of metric spaces

The class of metric spaces (X,d) known as small-determined spaces, introduced by Garrido and Jaramillo, are properly defined by means of some type of real-valued Lipschitz functions on X. On the other hand, B-simple metric spaces introduced by Hejcman are defined in terms of some kind of bornologies of bounded subsets of X. In this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties.

Linear non-local diffusion problems in metric measure spaces

The aim of this paper is to provide a comprehensive study of some linear non-local diffusion problems in metric measure spaces. These include, for example, open subsets in ℝN, graphs, manifolds, multi-structures and some fractal sets. For this, we study regularity, compactness, positivity and the spectrum of the stationary non-local operator. We then study the solutions of linear evolution non-local diffusion problems, with emphasis on similarities and differences with the standard heat equation in smooth domains. In particular, we prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods.