Gradient flows in asymmetric metric spaces

Abstract

This article is concerned with gradient flows in asymmetric metric spaces, that is, spaces with a topology induced by an asymmetric metric. Such an asymmetry appears naturally in many applications, e.g., in mathematical models for materials with hysteresis. A framework of asymmetric gradient flows is established under the assumption that the metric is weakly lower-semicontinuous in the second argument (and not necessarily on the first), and an existence theorem for gradient flows defined on an asymmetric metric space is given.