04140nam a22005655i 4500 978-3-031-51881-2 DE-He213 20251030112053.0 cr nn 008mamaa 240522s2024 sz | s |||| 0|eng d 9783031518812 978-3-031-51881-2 10.1007/978-3-031-51881-2 doi QA402.5-402.6 QA315-316 PBU bicssc MAT042000 bisacsh PBU thema Górny, Wojciech. author. aut http://id.loc.gov/vocabulary/relators/aut 77216 Functions of Least Gradient [electronic resource] / by Wojciech Górny, José M. Mazón. 1st ed. 2024. Cham : Springer Nature Switzerland : Imprint: Birkhäuser, 2024. XXVIII, 428 p. 20 illus., 9 illus. in color. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Monographs in Mathematics, 2296-4886 ; 110 Least Gradient Problem and Minimal Surfaces -- Existence and Characterization of Weak Solutions -- Anisotropic Least Gradient Problem -- Duality Approach -- Existence of Solutions in the Trace Sense -- Uniqueness and Structure of Solutions -- Regularity of Solutions. 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. Mathematical optimization. 65522 Calculus of variations. 67268 Differential equations. Calculus of Variations and Optimization. 67270 Differential Equations. Mazón, José M. author. aut http://id.loc.gov/vocabulary/relators/aut 77217 SpringerLink (Online service) Springer Nature eBook Printed edition: 9783031518805 Printed edition: 9783031518829 Printed edition: 9783031518836 Monographs in Mathematics, 2296-4886 ; 110 77218 https://doi.org/10.1007/978-3-031-51881-2 ZDB-2-SMA ZDB-2-SXMS 60027 60027