Curves and Gradients in Metric Spaces

Abstract

As we briefly discussed in the introduction, the notion of gradient flows in a metric space relies on two elementary but basic concepts: the metric derivative of an absolutely continuous curve with values in and the upper gradients of a functional defined in . The related definitions are presented in the next two sections (a more detailed treatment of this topic can be found for instance in [20]); the last one deals with curves of maximal slope.