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Birkhäuser

Functions of Least Gradient

  • Book
  • May 2024
  • Latest edition

Overview

Authors:
  1. Wojciech Górny

    1. Faculty of Mathematics, University of Vienna, Vienna, Austria

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    You can also search for this author in PubMed   Google Scholar

  2. José M. Mazón

    1. Department of Mathematical Analysis, University of Valencia, Burjassot, Spain

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Stands out as the first book on least gradient functions
  • Presents both classic and new results on this topic
  • Features open problems

Part of the book series: Monographs in Mathematics (MMA, volume 110)

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Keywords

  • Least Gradient Functions
  • Functions of Bounded Variation
  • Area-minimizing Sets
  • Minimal Surface
  • Anisotropy
  • 1-Laplacian

About this book

This book is devoted to the least gradient problem and its variants. The least gradient problem concerns minimization of the total variation of a function with prescribed values on the boundary of a Lipschitz domain. It is the model problem for studying minimization problems involving functionals with linear growth. Functions which solve the least gradient problem for their own boundary data, which arise naturally in the study of minimal surfaces, are called functions of least gradient.

The main part of the book is dedicated to presenting the recent advances in this theory. Among others are presented an Euler–Lagrange characterization of least gradient functions, an anisotropic counterpart of the least gradient problem motivated by an inverse problem in medical imaging, and state-of-the-art results concerning existence, regularity, and structure of solutions. Moreover, the authors present a surprising connection between the least gradient problem and the Monge–Kantorovich optimal transport problem and some of its consequences, and discuss formulations of the least gradient problem in the nonlocal and metric settings. Each chapter is followed by a discussion section concerning other research directions, generalizations of presented results, and presentation of some open problems.



The book is intended as an introduction to the theory of least gradient functions and a reference tool for a general audience in analysis and PDEs. The readers are assumed to have a basic understanding of functional analysis and partial differential equations. Apart from this, the text is self-contained, and the book ends with five appendices on functions of bounded variation, geometric measure theory, convex analysis, optimal transport, and analysis in metric spaces.

Authors and Affiliations

  • Faculty of Mathematics, University of Vienna, Vienna, Austria

    Wojciech Górny

  • Department of Mathematical Analysis, University of Valencia, Burjassot, Spain

    José M. Mazón

About the authors

Wojciech Górny graduated from the University of Warsaw. Currently, he is a senior postdoc at the University of Vienna. He works primarily in calculus of variations, functional analysis, and partial differential equations.

José M. Mazón is a professor emeritus of the Department of Mathematical Analysis at the University of Valencia. His main field of research are nonlinear partial differential equations.

Bibliographic Information

  • Book Title: Functions of Least Gradient

  • Authors: Wojciech Górny, José M. Mazón

  • Series Title: Monographs in Mathematics

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024

  • Hardcover ISBN: 978-3-031-51880-5Due: 20 May 2024

  • Softcover ISBN: 978-3-031-51883-6Due: 20 May 2024

  • eBook ISBN: 978-3-031-51881-2Due: 20 May 2024

  • Series ISSN: 1017-0480

  • Series E-ISSN: 2296-4886

  • Edition Number: 1

  • Number of Pages: XXVIII, 424

  • Number of Illustrations: 11 b/w illustrations, 9 illustrations in colour

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