Heat flow and boundary value problem for harmonic maps
type: Item
instance of: scholarly article
title: Heat flow and boundary value problem for harmonic maps
zbMATH Open document ID: 0687.58004
DOI: 10.1016/S0294-1449(16)30316-X
author: Kung Ching Chang
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1989-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78184
review text: The paper is concerned with the existence of global solutions \(f: [0,+\infty[ \times M \to N\) to the parabolic system \[ \begin{alignedat}{2} \partial_ t f(t,x) &= \underline\Delta f(t,x) &\qquad&\text{in }[0,+\infty[\times M,\\ f(0,x) &= \varphi(x) &\qquad&\text{on }M,\\ f(t,x) &= \chi(x) &\qquad&\text{on }[0,+\infty[ \times \partial M, \end{alignedat} \] where \(M\) and \(N\) are given Riemannian manifolds and \(\varphi : M \to N\) and \(\chi : \partial M \to N\) are assigned maps with \(\varphi|_{\partial M} = \chi\). The main result asserts the existence of a classical solution f, provided that the energy \(\frac12 \int_{M} |\nabla\varphi|^2 dV\) is sufficiently small. Moreover, along some sequence \((t_ j) \to +\infty\), \(f(t_ j,\cdot)\) is convergent to some harmonic map \(\tilde u\) with \(\tilde u |_{\partial M} = \chi\). As a consequence, some known results, concerning the existence of harmonic maps, are re-obtained. Also a result of Ljusternik-Schnirelmann type is given.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 4126339
zbMATH Keywords: heat equation
zbMATH Keywords: minimax principle
zbMATH Keywords: parabolic system
zbMATH Keywords: harmonic map
zbMATH Keywords: Ljusternik-Schnirelmann
reviewed by: Marco Degiovanni
MaRDI profile type: Publication
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OpenAlex ID: W2267987739
description: scientific article; zbMATH DE number 4126339