Higher order energy functionals

Branding, Volker; Montaldo, Stefano; Oniciuc, Cezar; Ratto, Andrea

doi:10.1016/j.aim.2020.107236

Cited by 31 publications

(46 citation statements)

References 26 publications

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“…MJOM mapping [7], suggested the idea of studying critical points of higher order energies as a possible generalization of harmonic maps. For an updated account on higher order energies we recommend the interested reader to see [4].…”

Section: S Montaldo and A Pámpanomentioning

confidence: 99%

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Triharmonic Curves in 3-Dimensional Homogeneous Spaces

Montaldo

Pámpano

2021

Mediterr. J. Math.

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We first prove that, unlike the biharmonic case, there exist triharmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classification of triharmonic curves in surfaces with constant Gaussian curvature. Next, restricting to curves in a 3-dimensional Riemannian manifold, we study the family of triharmonic curves with constant curvature, showing that they are Frenet helices. In the last part, we give the full classification of triharmonic Frenet helices in space forms and in Bianchi–Cartan–Vranceanu spaces.

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Section: S Montaldo and A Pámpanomentioning

confidence: 99%

“…For a complete description of the relations between the functional (1) and the functionals (2) and (3), we refer the reader to [4].…”

Section: Triharmonic Curves In Riemannian Manifoldsmentioning

confidence: 99%

Triharmonic Curves in 3-Dimensional Homogeneous Spaces

Montaldo

Pámpano

2021

Mediterr. J. Math.

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“…The study of these functionals was proposed by Eells and Sampson in 1965 (see [11]) and, later, by Eells and Lemaire in 1983 [9, p.77, Problem (8.7)]. A rigorous mathematical investigation of (1.9) has recently been initiated by the authors in [6] and was further developed in [5,20]. We point out that, in general, the functional (1.9) coincides with E k (ϕ) introduced above only when k = 1, 2, 3.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…In this section we study unique continuation properties for critical points of the ES-4-energy (1.9). We shall prove two unique continuation results for k = 4, which is the only case for which we know the explicit form of the Euler-Lagrange equations (see [6]). The energy functional for ES-4-harmonic maps (corresponding to (1.9) with k = 4) is given by and the termτ 4 (ϕ) is defined bŷ…”

Section: Unique Continuation Theorems For Es-4-harmonic Mapsmentioning

confidence: 93%

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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“…The first variation of (1.10) was calculated in [3,Section 3] and is characterized by the vanishing of the ES-4-tension field τ ES 4 (ϕ) given by the following expression…”

Section: Introduction and Resultsmentioning

confidence: 99%

On Finite Energy Solutions of 4-harmonic and ES-4-harmonic Maps

Branding

2021

J Geom Anal

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Abstract4-harmonic and ES-4-harmonic maps are two generalizations of the well-studied harmonic map equation which are both given by a nonlinear elliptic partial differential equation of order eight. Due to the large number of derivatives it is very difficult to find any difference in the qualitative behavior of these two variational problems. In this article we prove that finite energy solutions of both 4-harmonic and ES-4-harmonic maps from Euclidean space must be trivial. However, the energy that we require to be finite is different for 4-harmonic and ES-4-harmonic maps pointing out a first difference between these two variational problems.

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Higher order energy functionals

Cited by 31 publications

References 26 publications

Triharmonic Curves in 3-Dimensional Homogeneous Spaces

Triharmonic Curves in 3-Dimensional Homogeneous Spaces

Unique continuation properties for polyharmonic maps between Riemannian manifolds

On Finite Energy Solutions of 4-harmonic and ES-4-harmonic Maps

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