Overview

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 Euler–Lagrange characterisation of least gradient functions in this setting, and how to study metric counterparts of Cheeger problems.

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9781009741125

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Author Information

Wojciech Górny is currently Associate Professor at the University of Warsaw and works primarily in calculus of variations, functional analysis, and PDEs. He previously co-authored the book 'Functions of Least Gradient' (2024) with Jose M. Mazón. José M. Mazón is Professor Emeritus in the Department of Mathematical Analysis at the University of Valencia. His research centres around Nonlinear Partial Differential Equations. He has authored approximately 130 papers and five books, one of which was awarded the Ferran Sunyer i Balaguer Prize in 2003.

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