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Filling a gap in the literature, this book explores the theory of gradient flows of convex functionals in metric measure spaces, with an emphasis on weak solutions. It is largely self-contained and assumes only a basic understanding of functional analysis and partial differential equations.

With appendices on convex analysis and the basics of analysis in metric spaces, it provides a clear introduction to the topic for graduate students and non-specialist researchers, and a useful reference for anyone working in analysis and PDEs.

The text focuses on several key recent developments and advances in the field, paying careful attention to technical detail. These include how to use a first-order differential structure to construct weak solutions to the p-Laplacian evolution equation and the total variation flow in metric spaces, how to show a EulerLagrange characterisation of least gradient functions in this setting, and how to study metric counterparts of Cheeger problems.