Existence of Equivariant Biharmonic Maps

Hornung, Peter; Moser, Roger

doi:10.1093/imrn/rnv212

Cited by 5 publications

(4 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Equivariant theory deals with special families of maps having enough symmetries to guarantee that harmonicity reduces to the study of a second order ordinary differential equation (we refer to [1,5,10,18,20,22] for background and examples). In [18], we developped a systematic approach to equivariant theory for biharmonic maps (see [15], for recent developments). In this framework, the aim of this work is to study rotationally symmetric biharmonic maps between two m-dimensional models (in the sense of [11]).…”

Section: Introductionmentioning

confidence: 99%

Rotationally symmetric biharmonic maps between models

Montaldo

Oniciuc

Ratto

2015

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Abstract. The main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two m-dimensional models (in the sense of Greene and Wu). We obtain a complete classification of rotationally symmetric, proper biharmonic conformal diffeomorphisms in the special case that m = 4 and the models have constant sectional curvature. Then, by introducing the Hamiltonian associated to this problem, we also obtain a complete description of conformal proper biharmonic solutions in the case that the domain model is R 4 . In the second part of the paper we carry out a stability study with respect to equivariant variations (equivariant stability). In particular, we prove that: (i) the inverse of the stereographic projection from the open 4-dimensional Euclidean ball to the hyperbolic space is equivariant stable; (ii) the inverse of the stereographic projection from the closed 4-dimensional Euclidean ball to the sphere is equivariant stable with respect to variations which preserve the boundary data.

show abstract

Section: Introductionmentioning

confidence: 99%

Rotationally symmetric biharmonic maps between models

Montaldo

Oniciuc

Ratto

2015

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…The auxiliary results contained in this section will be used later in the present paper, when discussing the asymptotic behavior of LdG minimizers at isolated singularities, and they will be also of use in our companion paper [12]. Since such profiles will be described in terms of S 1equivariant harmonic maps, these classification results are of independent interest and of possible use in the analysis of minimizing harmonic maps under simmetry constraint (see, e.g., [15,21] and references therein).…”

Section: Equivariant Harmonic Spheres Into Smentioning

confidence: 99%

“…We shall prove in Proposition 6.1 that a critical point in the symmetric class W 1,2 sym (Ω; S 4 ) is always a critical point in the global class W 1,2 (Ω; S 4 ) in the spirit of the general Palais symmetric criticality principle [38]. Note that this principle does not directly apply here since W 1,2 (Ω; S 4 ) and W 1,2 sym (Ω; S 4 ) are not Banach manifolds, and we need to prove it by hands (see also [15] and [21] for a similar results in the context of harmonic and biharmonic maps respectively).…”

Section: Partial Regularity Of Ldg Minimizers Under Axial Symmetrymentioning

confidence: 99%

Torus-like solutions for the Landau-de Gennes model. Part II: Topology of $\mathbb{S}^1$-equivariant minimizers

Dipasquale,

Millot,

Pisante

2020

Preprint

View full text Add to dashboard Cite

We study energy minimization of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional axisymmetric domains and in a restricted class of S 1 -equivariant configurations. We assume smooth and nonvanishing S 1 -equivariant (e.g. homeotropic) Dirichlet boundary conditions and a physically relevant norm constraint (Lyuksyutov constraint) in the interior. Relying on results in [11] in the nonsymmetric setting, we prove partial regularity of minimizers away from a possible finite set of interior singularities lying on the symmetry axis. For a suitable class of domains and boundary data we show that for smooth minimizers (torus solutions) the level sets of the signed biaxiality are generically finite union of tori of revolution. In case of a nematic droplet, we provide existence of torus solutions, at least when the boundary data are suitably chosen continuous deformations of the radial anchoring. Concerning nonsmooth minimizers (split solutions), we characterize explicitely their asymptotic behavior around any singular point in terms of explicit S 1 -equivariant harmonic maps into S 4 , whence the generic level sets of the signed biaxiality contains invariant topological spheres. In the companion paper [12], we will show how singular solutions or smooth solutions (or even both) for the Euler-Lagrange equations do appear as minimizers under radial anchoring boundary data when the domains are suitable deformation of a spherical droplet.

show abstract

“…In this section we study rotationally symmetric maps between models, as defined in (2.12). Further details on this topic and related examples can be found in [7], [31], [43], [44] and [52]. First, by using (2.5), (2.15) and (2.19), we easily compute the tension field of a rotationally symmetric map as in (2.12).…”

Section: Rotationally Symmetric Biharmonic Maps Between Modelsmentioning

confidence: 99%

Reduction methods for the bienergy

Montaldo,

Oniciuc,

Ratto

2016

Preprint

View full text Add to dashboard Cite

This paper, in which we develop ideas introduced in [44], focuses on reduction methods (basically, group actions or, more generally, simmetries) for the bienergy. This type of techniques enable us to produce examples of critical points of the bienergy by reducing the study of the relevant fourth order PDE's system to ODE's. In particular, we shall study rotationally symmetric biharmonic conformal diffeomorphisms between models. Next, we will adapt the reduction method to study an ample class of G−invariant immersions into the Euclidean space. At present, the known instances in these contexts are far from reaching the depth and variety of their companions which have provided fundamental solutions to classical problems in the theories of harmonic maps and minimal immersions. However, we think that these examples represent an important starting point which can inspire further research on biharmonicity. In this order of ideas, we end this paper with a discussion of some open problems and possible directions for further developments.

show abstract

Existence of Equivariant Biharmonic Maps

Abstract: We consider two compact Riemannian manifolds M and N and a compact Lie group G that acts on both by isometries. Under certain assumptions on the structure of M and of the quotient space M/G, we construct equivariant biharmonic maps u : M → N with prescribed boundary data.

Cited by 5 publications

References 20 publications

Rotationally symmetric biharmonic maps between models

Rotationally symmetric biharmonic maps between models

Torus-like solutions for the Landau-de Gennes model. Part II: Topology of $\mathbb{S}^1$-equivariant minimizers

Reduction methods for the bienergy

Contact Info

Product

Resources

About