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Unique continuation properties for polyharmonic maps between Riemannian manifolds

Part of: Variational problems in infinite-dimensional spaces Global differential geometry Higher-dimensional theory

Published online by Cambridge University Press:  25 August 2021

Volker Branding ,
Stefano Montaldo ,
Cezar Oniciuc  and
Andrea Ratto
Volker Branding*
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
Stefano Montaldo
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy e-mail: montaldo@unica.it rattoa@unica.it
Cezar Oniciuc
Affiliation:
Faculty of Mathematics, Al.I. Cuza University of Iasi, Bulevardul Carol I, 11, 700506 Iasi, Romania e-mail: oniciucc@uaic.ro
Andrea Ratto
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy e-mail: montaldo@unica.it rattoa@unica.it
*
e-mail: volker.branding@univie.ac.at
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Abstract

Polyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher-order functionals which extend the classical energy functional for maps between Riemannian manifolds. The main aim of this paper is to investigate the so-called unique continuation principle. More precisely, assuming that the domain is connected, we shall prove the following extensions of results known in the harmonic and biharmonic cases: (i) if a k-harmonic map is harmonic on an open subset, then it is harmonic everywhere; (ii) if two k-harmonic maps agree on an open subset, then they agree everywhere; and (iii) if, for a k-harmonic map to the n-dimensional sphere, an open subset of the domain is mapped into the equator, then all the domain is mapped into the equator.

Keywords

polyharmonic mapsunique continuation principlehigher-order elliptic operators

MSC classification

Primary: 58E20: Harmonic maps , etc. 31B30: Biharmonic and polyharmonic equations and functions 53C20: Global Riemannian geometry, including pinching
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 1 , February 2023 , pp. 1 - 28
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

The first author gratefully acknowledges the support of the Austrian Science Fund (FWF) through the START-project Y963-N35 of Michael Eichmair and the project P30749-N35 “Geometric variational problems from string theory”. The second author and the last author are members of the GNSAGA group of INdAM (Italy) and they are also supported by the Fondazione di Sardegna (project STAGE) and Regione Autonoma della Sardegna (Project KASBA). The third author was partially supported from the grant PN-III-P4-ID-PCE-2020-0794.

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