LEADER 03785nam 22005655 450 001 9910864186003321 005 20250807153146.0 010 $a3-031-51881-0 024 7 $a10.1007/978-3-031-51881-2 035 $a(CKB)32138222200041 035 $a(MiAaPQ)EBC31352131 035 $a(Au-PeEL)EBL31352131 035 $a(DE-He213)978-3-031-51881-2 035 $a(OCoLC)1435754522 035 $a(EXLCZ)9932138222200041 100 $a20240522d2024 u| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aFunctions of Least Gradient /$fby Wojciech Górny, José M. Mazón 205 $a1st ed. 2024. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Birkhäuser,$d2024. 215 $a1 online resource (442 pages) 225 1 $aMonographs in Mathematics,$x2296-4886 ;$v110 311 08$a3-031-51880-2 327 $aLeast 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. 330 $aThis book is devoted to the least gradient problem and its variants. The least gradient problem concerns minimization of the total variation of a function with prescribed values on the boundary of a Lipschitz domain. It is the model problem for studying minimization problems involving functionals with linear growth. Functions which solve the least gradient problem for their own boundary data, which arise naturally in the study of minimal surfaces, are called functions of least gradient. The main part of the book is dedicated to presenting the recent advances in this theory. Among others are presented an Euler?Lagrange characterization of least gradient functions, an anisotropic counterpart of the least gradient problem motivated by an inverse problem in medical imaging, and state-of-the-art results concerning existence, regularity, and structure of solutions. Moreover, the authors present a surprising connection between the least gradient problem and the Monge?Kantorovich optimal transport problem and some of its consequences, and discuss formulations of the least gradient problem in the nonlocal and metric settings. Each chapter is followed by a discussion section concerning other research directions, generalizations of presented results, and presentation of some open problems. The book is intended as an introduction to the theory of least gradient functions and a reference tool for a general audience in analysis and PDEs. The readers are assumed to have a basic understanding of functional analysis and partial differential equations. Apart from this, the text is self-contained, and the book ends with five appendices on functions of bounded variation, geometric measure theory, convex analysis, optimal transport, and analysis in metric spaces. 410 0$aMonographs in Mathematics,$x2296-4886 ;$v110 606 $aMathematical optimization 606 $aCalculus of variations 606 $aDifferential equations 606 $aCalculus of Variations and Optimization 606 $aDifferential Equations 615 0$aMathematical optimization. 615 0$aCalculus of variations. 615 0$aDifferential equations. 615 14$aCalculus of Variations and Optimization. 615 24$aDifferential Equations. 676 $a515.7 700 $aGo?rny$b Wojciech$01740647 701 $aMazón$b José M$0421392 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910864186003321 996 $aFunctions of Least Gradient$94166426 997 $aUNINA