Regularity for the evolution of \(p\)-harmonic maps
type: Item
instance of: scholarly article
title: Regularity for the evolution of \(p\)-harmonic maps
zbMATH Open document ID: 1400.35173
author: Masashi Misawa
published in: Journal of Differential Equations
publication date: 2017-11-14T00:00:00Z
review text: The author studies the regularity for \(p\)-harmonic map heat flows. A \(p\)-harmonic map is defined to be a critical point, which is a solution of the Euler-Lagrange equation of the \(p\)-energy \(-\mathrm{div}(\text{ }\!\!|\!\!\text{ Du}{{\text{ }\!\!|\!\!\text{ }}^{\text{p-2}}}\text{Du)= }\!\!|\!\!\text{ Du}{{\text{ }\!\!|\!\!\text{ }}^{\text{p-2}}}A(u)(\text{Du},\text{Du})\), where \(\Omega \subset {{\mathbb{R}}^{m}},\;m\geq 2\) be a bounded domain with a smooth boundary, \(Nu\) is an \(n\)-dimensional smooth compact Riemannian manifold without a boundary assumed to be isometrically embedded in \({{\mathbb{R}}^{l}}\), \(l>n\), \(u:\Omega \to \text{N}\), \(\text{Du}=\mathrm{grad}(u)\), \(A(\cdot )(\cdot ,\cdot )=(\sum\limits_{j,k=1}^{l}{A_{jk}^{i}}(u)\text{D}{{\text{u}}^{j}}\cdot \text{D}{{\text{u}}^{k}}),\;i=1,\dots,l\) is the second fundamental form of \(N \subset {{\mathbb{R}}^{l}}\). It is devised a monotonicity type formula of scaled energy and established a criterion for a uniform regularity estimate for the regular \(p\)-harmonic map heat flows. In additional as application of the obtained results are given the small data global in the time existence of regular \(p\)-harmonic map heat flow.
reviewed by: Andrey Zahariev
Mathematics Subject Classification ID: 35K92
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 6806982
zbMATH Keywords: \(p\)-Laplacian system
zbMATH Keywords: \(p\)-harmonic map heat flow
zbMATH Keywords: \(p\)-harmonic map
zbMATH Keywords: regularity
zbMATH Keywords: monotonicity estimate
MaRDI profile type: Publication
full work available at URL: j.jde.2017.10.006
OpenAlex ID: W2765230275
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DOI: 10.1016/J.JDE.2017.10.006
description: scientific article; zbMATH DE number 6806982