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# Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

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Volume 11, Issue 3

# Local regularity and compactness for the p-harmonic map heat flows

Masashi Misawa
• Corresponding author
• Department of Mathematics, Faculty of Sciences, Kumamoto University,2-39-1 Kurokami, Kumamoto-shi, Kumamoto 860-8555, Japan
• Email
• Other articles by this author:
Published Online: 2017-02-21 | DOI: https://doi.org/10.1515/acv-2016-0064

## Abstract

We study a geometric analysis and local regularity for the evolution of $p$-harmonic maps, called $p$-harmonic map heat flows. Our main result is to establish a criterion for a uniform local regularity estimate for regular $p$-harmonic map heat flows, devising some new monotonicity-type formulas of a local scaled energy. The regularity criterion obtained is almost optimal, comparing with that of the corresponding stationary case. As application we show a compactness of regular $p$-harmonic map heat flows with energy bound.

Keywords: local regularity; monotonicity estimates

MSC 2010: 35B45; 35B65; 35D30; 35K59; 35K65

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M. Misawa, Regularity for the evolution of $p$-harmonic maps, to appear.

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Accepted: 2017-01-17

Published Online: 2017-02-21

Published in Print: 2018-07-01

Funding Source: Japan Society for the Promotion of Science

Award identifier / Grant number: 15K04962

The work of Masashi Misawa was partially supported by the Grant-in-Aid for Scientific Research (C) No. 15K04962 from Japan Society for the Promotion of Science.

Citation Information: Advances in Calculus of Variations, Volume 11, Issue 3, Pages 223–255, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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