Nonlinear Markov operators associated with symmetric Markov kernels and energy minimizing maps between singular spaces
type: Item
instance of: scholarly article
title: Nonlinear Markov operators associated with symmetric Markov kernels and energy minimizing maps between singular spaces
zbMATH Open document ID: 0987.31006
author: Karl-Theodor Sturm
published in: Calculus of Variations and Partial Differential Equations
publication date: 2002-06-16T00:00:00Z
review text: The author considers harmonic maps defined on a space with measure \(m\) and a symmetric Markov kernel \(p\) on it, with values in a complete geodesic space of non-positive curvature in the sense of A. D. Alexandrov. Thus, the theme of the paper belongs to the theory of nonlinear Dirichlet forms and energy minimizing maps between singular spaces as developed, e.g. by \textit{J. Jost} [Calc. Var. Partial Differ. Equ. 2, 173-204 (1994; Zbl 0798.58021)] and \textit{N. Korevaar} and \textit{R. Schoen} [Commun. Anal. Geom. 1, 561-659 (1993; Zbl 0862.58004)]. It is proved in fact that the harmonic maps are exactly the minimizers of the energy. The paper specializes (to the case where \(p\) is symmetric with respect to the measure \(m\)) results obtained previously by the author: Nonlinear Markov operators, discrete heat flow, and harmonic maps between singular spaces, Potential Anal. 16, No. 4, 305-340 (2002). As such, it may be considered as a nonlinear generalization of the theory of symmetric Markov kernels; various properties of the linear heat flow carry over to the nonlinear case.
reviewed by: Liliana Popa
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 60J45
zbMATH DE Number: 1655599
zbMATH Keywords: harmonic map
zbMATH Keywords: nonlinear Markov operator
zbMATH Keywords: nonlinear heat flow
zbMATH Keywords: nonlinear Dirichlet form
MaRDI profile type: Publication
DOI: 10.1007/S005260000057
description: scientific article; zbMATH DE number 1655599