Unique continuation theorems for biharmonic maps

Branding, Volker; Oniciuc, Cezar

doi:10.1112/blms.12240

Cited by 17 publications

(22 citation statements)

References 12 publications

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“…So far, only few results on interpolating sesqui-harmonic maps have been established. Besides a number of general features [5] the unique continuation property for interpolating sesquiharmonic maps has been proved in [9]. In this article we will mostly focus on analytic aspects of interpolating sesqui-harmonic maps building on the regularity theory developed for biharmonic maps [10,16].…”

Section: Introduction and Resultsmentioning

confidence: 99%

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Some analytic results on interpolating sesqui-harmonic maps

Branding

2020

Annali di Matematica

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In this article we study various analytic aspects of interpolating sesqui-harmonic maps between Riemannian manifolds where we mostly focus on the case of a spherical target. The latter are critical points of an energy functional that interpolates between the functionals for harmonic and biharmonic maps. In the case of a spherical target we will derive a conservation law and use it to show the smoothness of weak solutions. Moreover, we will obtain several classification results for interpolating sesqui-harmonic maps.Date: July 10, 2019. 2010 Mathematics Subject Classification. 58E20; 31B30; 35B65. Key words and phrases. interpolating sesqui-harmonic maps; regularity of weak solutions; classification results.Applying Proposition 2.10, choosing λ = 4, p * = 4 and α = 1, we find.Applying Proposition 2.10 again we find by choosing p = 4 3 , α = 2 and λ = 4 that).Using Hölder's inequality we get the estimates |∆φ||∇φ| M 4 3 ,4 (R m ) ≤ ∆φ M 2,4 (R m ) ∇φ M 4,4 (R m ) , |∇φ| 3 M 4 3 ,4 (R m ) ≤ ∇φ 3 M 4,4 (R m ) and also ∇φ M 4 3 ,4 (R m )

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Section: Introduction and Resultsmentioning

confidence: 99%

“…So far, only few results on interpolating sesqui-harmonic maps have been established. Besides a number of general features [ 5 ], the unique continuation property for interpolating sesqui-harmonic maps has been proved in [ 9 ].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Some analytic results on interpolating sesqui-harmonic maps

Branding

2020

Annali di Matematica

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“…First of all, thanks to the unique continuation theorem for biharmonic maps (see [5]) which states that two biharmonic maps are the same if they agree on an open subset of the domain, it is enough to prove the classification theorem in an open subset of the domain.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…Applying ∇ to both sides of ( 6) and adding the results to (5) gives the following necessary condition for a conformal biharmonic maps between two space forms.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

Some classifications of conformal biharmonic and k-polyharmonic maps

Ou¹

2021

Preprint

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We give a complete classification of local and global conformal biharmonic maps between any two space forms by proving that a conformal map between two space forms is proper biharmonic if and only if the dimension is 4, the domain is flat, and it is a restriction of a Möbius transformation. We also show that proper k-polyharmonic conformal maps between Euclidean spaces exist if and only if the dimension is 2k and they are precisely the restrictions of Möbius transformations. This provides infinitely many simple examples of proper k-polyharmonic maps with nice geometric structure.

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“…7),(1.8), denoted by F k . Note that this is the only place in(2.23) where the Euler-Lagrange equation for polyharmonic maps enters.…”

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Unique continuation properties for polyharmonic maps between Riemannian manifolds

Branding

Montaldo²,

Oniciuc³

et al. 2021

Can. J. Math.-J. Can. Math.

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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 case: (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 a open subset, then they agree everywhere; (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.

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Unique continuation theorems for biharmonic maps

Abstract: We prove several unique continuation results for biharmonic maps between Riemannian manifolds.

Cited by 17 publications

References 12 publications

Some analytic results on interpolating sesqui-harmonic maps

Some analytic results on interpolating sesqui-harmonic maps

Some classifications of conformal biharmonic and k-polyharmonic maps

Unique continuation properties for polyharmonic maps between Riemannian manifolds

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